Existence, uniqueness and regularity of solutions to the stochastic Landau–Lifshitz–Slonczewski equation

Abstract

In this paper we are concerned with the stochastic Landau–Lifshitz–Slonczewski (LLS) equation that describes magnetisation of an infinite nanowire evolving under current-driven spin torque. The current brings into the system a multiplicative gradient noise that appears as a transport term in the equation. We prove the existence, uniqueness and regularity of pathwise solutions to the equation.

1 Introduction

In this paper we are concerned with the existence, uniqueness and regularity of solutions to the stochastic Landau–Lifshitz–Slonczewski (LLS) equation considered on real line, see (2). This is a result on a system of stochastic PDEs that combines variational structure with the transport noise in the presence of geometric constraints, more precisely, with solutions taking values in a sphere.

Let us recall briefly the physical motivation for the LLS equation, see [9, Chapter 9] for more details. In a nanoscale ferromagnetic element, the interaction between an electric current and the magnetisation of the element can result in current-induced magnetisation switching and spin wave emission. This observation gave rise to the rapidly developing field of spintronics, including racetrack memory [31] and microwave oscillators [24]. It is expected that a good understanding of the current-induced magnetisation switching and the stability of the domain walls subject to random perturbations will allow us to develop new types of current-controlled magnetic memories.

Slonczewski [38] first modelled the magnetisation dynamics in a magnetic multilayer structure with the current flowing perpendicular to the layers. The first (fixed) layer is assumed to have a uniform magnetisation m that polarises the spins of the incoming current, and the spin accumulation $\textbf{s}$ entering the second (free) layer is uniform and parallel to m. This results in a spin transfer torque that affects the dynamics in the free layer. Viewing these layers as cross sections of a nanowire, and taking a continuous limit of the spin torque between infinitely thin cross sections [26, 40], we arrive at the LLS equation (1) with $\gamma = -\alpha$.

Zhang et al. [41] later proposed another model involving directly the time and spatially varying magnetisation m(t, x) and spin accumulation $\textbf{s}(t,x)$. The vector $\textbf{s}$ is still roughly parallel to m since the magnetisation varies at a much larger length scale than the Fermi wavelength of electron spins [23, 25], but the deviation of $\textbf{s}$ from m is non-trivial. Deriving from the generalised spin continuity equation, $\textbf{s}$ satisfies a diffusion equation with terms depending on m, and enters the classic Landau–Lifshitz–Gilbert equation as an additional source of effective field. This fully coupled system in dimension 1 is formulated in [35, 41] and later reduced to (1) in [42] by viewing $\textbf{s}$ as a first-order approximation of m. In particular, the final effective spin torque in [42] consists of (the continuous limit of) Slonczewski’s torque and an effective field contribution $m \times \partial _x m$, both weighed by the current velocity. After rescaling, this leads to the LLS equation (1) with $\gamma = \frac{\beta -\alpha }{1+\alpha \beta }$ for a constant $\beta \in {\mathbb {R}}$ that depends on the exchange coupling and spin relaxation times.

Mathematical theory of the LLS equation is at an early stage. For the coupled system in [41], the existence and uniqueness of solutions were analysed in [16, 19, 32], and a related optimal control problem was investigated in [5]. We mention that in these papers, the estimates of (approximations of) $\textbf{s}$ and m could be decoupled, with bounds depending only on the current density. In comparison, the LLS equation (1) has the gradient of m in place of $\textbf{s}$, which leads to differences in proving estimates, such as the way to control the length of approximations of m. The case, when the ferromagnetic material fills in a three-dimensional domain, was studied in an important paper [26], where the existence and uniqueness of solutions were proved and their regularity was studied. A physically important case of a nanowire is a subject of ongoing intense research in physics.

Mathematical analysis of general travelling domain walls and their stability was only recently initiated in [27, 33, 36].

The necessity to include random fluctuations (such as thermal noise) into the dynamics of magnetisation has been conjectured by physicists for many years (see for example [9, 10, 29]. In spintronics control of stability of the current-driven domain walls is the main obstacle for developing new generation of magnetic memories. The existence and uniqueness of solutions and numerical analysis for the Landau–Lifshitz equation without the Slonczewski term but including random thermal fluctuations were intensely studied in recent years, see, for instance, [2, 6, 7, 11, 20,21,22]. Some numerical schemes for the stochastic equation are closely related to their deterministic counterparts, which also attracted a lot of attention in the last two decades, see [3, 4, 8, 14] and references therein. In this paper, we will consider the LLS equation perturbed by transport instead of thermal noise. In the literature, transport noise has been studied for various models, for example, linear gradient noise in stochastic Navier–Stokes equation [28] which has a regularising effect [18], and locally Lipschitz gradient noise in stochastic geometric wave equation [13, 30] which satisfies the corresponding geometric constraint by residing in the tangent space to the target manifold. We refer the reader to [1, 34] and references therein for more examples, where global well-posedness is achieved for equations with locally Lipschitz or locally monotone coefficients under a coercivity (and sometimes stochastic parabolicity) condition on the leading order terms. In comparison, Eq. (2) lacks coercivity, and it has a nonlinearity $m \times \partial _{xx}m$ (the Schrödinger map) which is absent in the drift of wave equations.

To describe the problem in more detail, let us first recall briefly the deterministic LLS equation. We will identify an infinite nanowire made of ferromagnetic material with a real line ${\mathbb {R}}$ and will denote by $m(t,x)\in {\mathbb {R}}^3$ the magnetisation vector at a time $t\ge 0$ and at a point $x\in {\mathbb {R}}$. For temperatures below the Curie point the length |m(t, x)| of this vector is constant in (t, x) [10] and hence can be assumed equal to 1:

$$\begin{aligned} m: [0,\infty ) \times {\mathbb {R}} \rightarrow {\mathbb {S}}^2. \end{aligned}$$

The LLS equation proposed in [38] describes the dynamics of the magnetisation vector subject to the spin-velocity field (electric current):

$$\begin{aligned} v: (0,\infty ) \times {\mathbb {R}} \rightarrow {\mathbb {R}}. \end{aligned}$$

It takes the form

$$\begin{aligned} \partial _t m = -m\times \partial _{xx}m -\alpha m\times \left( m\times \partial _{xx}m \right) -v \partial _x m + \gamma m\times \left( v \partial _x m \right) , \end{aligned}$$

(1)

with $\alpha >0$ and $\gamma \in {\mathbb {R}}$.

The term $v \partial _x m$ is known as the adiabatic term and the non-adiabatic term is given by $\gamma m \times (v \partial _x m)$.

We will consider a version of Eq. (1) with the spin-velocity field perturbed by noise:

$$\begin{aligned} \textrm{d} m= & -m\times \left( \partial _{xx}m + \alpha m\times \partial _{xx}m\right) \textrm{d}t -\partial _x m \circ (v\,\textrm{d}t+\textrm{d}W)\nonumber \\ & + \left( m\times \gamma \partial _x m\right) \circ (v\,\textrm{d}t+\textrm{d}W), \end{aligned}$$

(2)

where W is an infinite-dimensional Wiener process taking values in an appropriate function space.

We emphasise that noise arises in Eq. (2) in a way very different from the way it appears in stochastic Landau–Lifshitz equations studied in [11, 12, 20, 21]. While in the aforementioned papers it is a thermal noise arising inside the magnetic domain and has bounded diffusion coefficient, in (2) it is a transport noise brought into the system by the electric current and has the gradient of the solution as a diffusion coefficient. Another noise term corresponding to the thermal noise might be included in the model below without any substantial difficulties. In order to simplify rather complicated technical arguments, we consider in this paper the transport noise only. Another novelty of our work is the presence of a “quadratic” diffusion coefficient in the noise term in (2). Therefore, we are able to study the physically important case when both adiabatic and non-adiabatic random transport terms are included. We note stochastic Landau–Lifshitz equations studied in recent years usually do not include the quadratic diffusion term despite the fact it naturally arises from formally including the noise term in the energy functional, and when included [12], a cutoff function is required for the estimates even in the case without gradient in the noise. Due to those more natural assumptions, analysis of Eq. (2) requires much more delicate arguments.

In this paper, we will show that for every initial condition $m_0$ with

$$\begin{aligned} \left| m_0(x)\right| =1,\quad \int _{\mathbb {R}}\left| \partial _x m_0 \right| ^2\,\textrm{d}x<\infty , \end{aligned}$$

(3)

there exists a unique pathwise and strong in PDEs sense solution to (2). We will use the observation made in [26] that under the constraint $|m(t,x)|=1$ for all $(t,x) \in (0,\infty ) \times {\mathbb {R}}$, we have

$$\begin{aligned}\partial _x m =-m\times \left( m\times \partial _x m\right) ;\end{aligned}$$

hence, Eq. (2) can be written in the form

$$\begin{aligned} \begin{aligned} \textrm{d}m&= -m \times \left( \partial _{xx}m + \alpha m \times \partial _{xx}m \right) \textrm{d}t + m \times \left( m \times \partial _x m + \gamma \partial _x m \right) \circ \left( v \,\textrm{d}t+\,\textrm{d}W\right) , \end{aligned} \end{aligned}$$

(4)

with $m(0) = m_0$ satisfying (3). Once we prove the existence of the sphere-valued solution m, Eq. (4) reduces to (2). We will assume that W is a Wiener process taking values in $H^2({\mathbb {R}})$ and will prove the existence and uniqueness of pathwise solutions to this equation, see Theorem 2.5 for details. Due to the presence of gradient noise of multiplicative type, we can prove this theorem only for the Wiener process with the covariance small enough. Smallness of the noise is usually necessary for stochastic PDEs with gradient noise, even in the case of linear equations, see, for example, Chapter 6 of [15]. For our equation, this is also evident from (13) when $\gamma =0$.

Let us comment on the proofs of Theorems 2.5 and 2.6. Following the approach of [26] we use lattice approximations that are well-known in physics literature. In [39] they were used to derive the equation for Schrödinger maps as limits of equations for discrete systems of interacting spins modelling the Heisenberg ferromagnet. This approach allows us to construct approximating solutions that satisfy the sphere constraint.

Then we obtain a set of uniform estimates for the approximate solutions. This step requires using quadratic interpolations and requires many new estimates.

Next, we follow compactness type argument to prove the existence of a limiting point that is a strong in PDEs sense solution to stochastic LLS equation. Finally, in Theorem 2.6 we show the uniqueness of pathwise solutions and use the Yamada–Watanabe theorem in the same way as in [11].

2 Semi-discrete scheme and the main result

2.1 Notation

2.1.1 Function spaces

Let $\rho _w(x) = (1+x^2)^{-w}$ for $w \ge 0$. Clearly,

$$\begin{aligned} \rho _w(x) \in (0,1], \quad |\rho _w'(x)| \le w \rho _w(x), \end{aligned}$$

(5)

and for $w > \frac{1}{2}$, $\int _{\mathbb {R}}\rho _w(x) \ \textrm{d}x<\infty$. For $p\in [1,\infty )$, define the weighted Lebesgue space ${\mathbb {L}}^p_w$ by

$$\begin{aligned} {\mathbb {L}}^p_w&= \left\{ f: {\mathbb {R}}\rightarrow {\mathbb {R}}^3;\, \int _{{\mathbb {R}}} |f(x)|^p \rho _w(x)\, \textrm{d}x < \infty \right\} . \end{aligned}$$

If $w=0$, then we will write ${\mathbb {L}}^p$ instead of ${\mathbb {L}}^p_0$. We will denote by ${\mathbb {H}}_w^1$ the Hilbert space

$$\begin{aligned}\mathbb H_w^1=\left\{ f\in \mathbb L_w^2;\, Df\in \mathbb L_w^2\right\} .\end{aligned}$$

Let $0<w_1 < w_2$. Then $\rho _{w_1} > \rho _{w_2}$ and the embeddings ${\mathbb {L}}^2 \hookrightarrow {\mathbb {L}}^2_{w_1} \hookrightarrow {\mathbb {L}}^2_{w_2}$ are continuous with

$$\begin{aligned} |f|_{{\mathbb {L}}^2_{w_2}} \le |f|_{{\mathbb {L}}^2_{w_1}} \le |f|_{{\mathbb {L}}^2}, \quad \forall f \in {\mathbb {L}}^2. \end{aligned}$$

Moreover, the embeddings

$$\begin{aligned} {\mathbb {H}}_{w_1}^1 \hookrightarrow {\mathbb {L}}^2_{w_2}\quad \text {and}\quad {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^2 \hookrightarrow {\mathbb {H}}_{w_2}^1\end{aligned}$$

are compact, where $\mathring{\mathbb H}^2$ stands for a standard homogeneous Sobolev space of functions $f:{\mathbb {R}}\rightarrow {\mathbb {R}}^3$ with weak derivatives $Df, D^2 f \in {\mathbb {L}}^2$. The Laplace operator $\Delta$ considered in $\mathbb L^2_w$ with the domain $\mathbb H^2_w$ is variational, and the operator $A_1 = I-\Delta$ is invertible. For $\beta > 0$, let $\mathbb H^\beta _w$ denote the domain of $A_1^{\beta /2}$ endowed with the norm $|\cdot |_{\mathbb H^{\beta }_w}:= |A_1^{\beta /2} \cdot |_{{\mathbb {L}}^2_w}$ and with dual space $\mathbb H^{-\beta }_w$.

2.1.2 Assumption and notation

Let W be an $H^2({\mathbb {R}})$-valued Wiener process with the covariance operator Q. Then there exists a complete orthonormal sequence $\left\{ f_j;\,j\ge 1\right\}$ of $H^2({\mathbb {R}})$ made of eigenvectors of Q, that is

$$\begin{aligned}Qf_j=q^2_jf_j,\quad q^2:=\sum _jq^2_j<\infty ,\end{aligned}$$

and then we have

$$\begin{aligned}W(t)=\sum _{j=1}^\infty q_jW_j(t)f_j.\end{aligned}$$

The following is a standing assumption for the rest of the paper, and it will not be enunciated again.

Assumption 2.1

Let $m_0$ satisfy (3) with $|m_0|_{{\mathbb {L}}^\infty \cap \mathring{{\mathbb {H}}}^1} \le K_0$ for some constant $K_0 >0$. Let

$$\begin{aligned} C_\kappa ^2:=\left| \sum _{j=1}^\infty q_j^2 \left( f_j^2 + (f_j^\prime )^2 + (f_j^{\prime \prime })^2 \right) \right| _{{\mathbb {L}}^\infty }<\infty , \end{aligned}$$

(6)

and $v \in {\mathcal {C}}([0,T]; H^1({\mathbb {R}}))$ with

$$\begin{aligned} C_v:=\mathop {\mathrm {ess\,sup}}\limits _{t \in {\mathbb {R}}_+} |v(t)|_{{\mathbb {L}}^\infty }<\infty . \end{aligned}$$

(7)

Define a function

$$\begin{aligned}\kappa ^2(x)=\sum _{j=1}^\infty q_j^2f_j^2(x),\quad x\in {\mathbb {R}}.\end{aligned}$$

Remark 2.2

(a) Assumption 2.1 yields

$$\begin{aligned}|\kappa |_{{\mathbb {L}}^\infty } \le C_\kappa ,\quad \textrm{and}\quad |\kappa \kappa ^\prime |_{{\mathbb {L}}^\infty } \le C_\kappa ^2.\end{aligned}$$

(b) Every $\mathbb H^2$-valued finite-dimensional Wiener process satisfies (6) provided $f_j^{\prime \prime } \in \mathbb L^\infty$ for $j\ge 1$.

The following notations will be used throughout the paper. Let $G: {\mathbb {L}}^\infty \cap \mathring{{\mathbb {H}}}^1\rightarrow {\mathbb {L}}^2$ be defined as

$$\begin{aligned}G(m)= m \times \left( m \times \partial _x m \right) + m\times \gamma \partial _x m.\end{aligned}$$

Let ${\mathcal {G}}(m):= G^\prime (m) (G(m))$,

which can be expressed as

$$\begin{aligned} {\mathcal {G}}(m)&= (\gamma ^2 - |m|^2) m \times \left( m \times \partial _{xx}m \right) -2 \gamma |m|^2 m \times \partial _{xx}m - \gamma ^2 \partial _x m \times \left( m \times \partial _x m \right) \\&\quad - \left| m \times \partial _x m\right| ^2 m + \gamma \langle m, \partial _x m \rangle m \times \partial _x m. \end{aligned}$$

In the rest of the paper, we consider Eq. (4) in its Itô form:

$$\begin{aligned} \textrm{d}m=\left( F(m)+ \frac{1}{2}S(m)\right) \, \textrm{d}t + G(m)\, \textrm{d}W, \quad m(0) = m_0. \end{aligned}$$

(8)

Here,

$$\begin{aligned} F(m)&:= m \times \left( m \times v \partial _x m \right) + \gamma m\times v \partial _x m - m \times \left( \partial _{xx} m + \alpha m \times \partial _{xx} m \right) \, \end{aligned}$$

and the Stratonovich correction term S(m) takes the form

$$\begin{aligned} S(m)&:=\kappa ^2{\mathcal {G}}(m)+ \kappa \kappa ^\prime \left[ m\times (m\times G(m))+ \gamma m\times G(m)\right] \\&= \kappa ^2 {\mathcal {G}}(m) + \kappa \kappa ^\prime \left[ (\gamma ^2-|m|^2) m \times \left( m \times \partial _x m \right) -2 \gamma |m|^2 m \times \partial _x m \right] . \end{aligned}$$

2.2 Semi-discrete scheme

2.2.1 Discrete operators and discrete spaces

Let ${\mathbb {Z}}_h = \{ x = kh: k \in {\mathbb {Z}}\}$ denote a discretisation of the real line of mesh size $h > 0$. For $u: {\mathbb {Z}}_h \rightarrow {\mathbb {R}}^3$, we write $u^{\pm }(x)$ for $u(x \pm h)$, and we introduce discrete gradient and discrete Laplace operators:

$$\begin{aligned} \begin{aligned} \partial ^{h} u = \frac{1}{h} \left( u^+ - u \right) \quad \text {and}\quad \Delta ^h u = \frac{1}{h} (\partial ^{h}u- \partial ^{h} u^-). \end{aligned} \end{aligned}$$

(9)

Let ${\mathbb {L}}_h^\infty$, ${\mathbb {L}}_h^p$, ${\mathbb {H}}_h^1$ and ${\mathbb {E}}_h:= {\mathbb {L}}_h^\infty \cap \mathring{\mathbb {H}}_h^1$ be discrete spaces equipped with respective norms:

$$\begin{aligned} |u|_{{\mathbb {L}}_h^\infty }&= \sup _{x \in {\mathbb {Z}}_h} |u(x)|, \quad |u|_{{\mathbb {L}}_h^p}^p = h \sum _{x \in {\mathbb {Z}}_h} |u(x)|^p, \\ |u |_{{\mathbb {E}}_h}^2&= | u |_{{\mathbb {L}}_h^\infty }^2 + |\partial ^{h} u|_{{\mathbb {L}}_h^2}^2, \quad |u|_{\mathbb H_h^1}^2 = |u|_{{\mathbb {L}}_h^2}^2+ |\partial ^{h} u|^2_{{\mathbb {L}}_h^2}, \end{aligned}$$

where $p\in [1,\infty )$.

We will say that $u:[0,T]\times \Omega \times \mathbb Z_h\rightarrow {\mathbb {R}}^3$ is an $\mathbb E_h$-valued progressively measurable process if for every $x\in \mathbb Z_h$ the process $u(\cdot ,x)$ is progressively measurable and for every $t \in [0,T]$,

$$\begin{aligned}|u(t)|_{\mathbb E_h}<\infty ,\quad {\mathbb {P}}\text {-a.s.}\end{aligned}$$

In particular, the process $\left\{ |u(t)|_{\mathbb E_h};\,t\ge 0\right\}$ is progressively measurable.

Let ${\mathcal {E}}_h$ denote the space of ${\mathbb {E}}_h$-valued progressively measurable processes, with norm

$$\begin{aligned} | u |_{{\mathcal {E}}_h} = \sup _{t \in [0,T]} \left( {\mathbb {E}}\left[ | u(t) |_{{\mathbb {E}}_h}^2 \right] \right) ^{\frac{1}{2}}. \end{aligned}$$

2.2.2 Discrete equation

For $u \in {\mathbb {E}}_h$, we define

$$\begin{aligned} \begin{aligned} F^h(u)&= u \times \left( u \times v \partial ^h u \right) + \gamma u \times v \partial ^h u - u \times \left( \Delta ^h u + \alpha u \times \Delta ^h u \right) \\ G^h(u)&= u \times \left( u \times \partial ^h u \right) + \gamma u \times \partial ^h u\\ S^h(u)&= {\mathcal {G}}^h_\kappa (u) + \kappa \kappa ' \left[ u\times (u\times G^h(u))+ \gamma u\times G^h(u)\right] , \end{aligned} \end{aligned}$$

(10)

where

$$\begin{aligned} {\mathcal {G}}^h_\kappa (u)&= \frac{1}{2}\left( (\kappa ^2)^- + \kappa ^2 \right) G^h_1(u) + \kappa ^2 G^h_2(u) + (\kappa ^2)^- G^h_3(u)\\ G^h_1(u)&= (\gamma ^2 - |u|^2) u \times \left( u \times \Delta ^h u \right) -2 \gamma |u|^2 u \times \Delta ^h u \\ G^h_2(u)&= - \gamma ^2 \partial ^h u \times \left( u \times \partial ^h u \right) - |u \times \partial ^h u|^2 u \\ G^h_3(u)&= 2\gamma \langle u, \partial ^h u^- \rangle u \times \partial ^h u\,. \end{aligned}$$

Fix a terminal time $T \in (0,\infty )$; we describe the semi-discrete scheme for (8) as a stochastic differential equation in the space ${\mathbb {E}}_h$:

$$\begin{aligned} \begin{aligned} \textrm{d}m^h(t)&= \left( F^h(m^h(t)) + \frac{1}{2} S^h(m^h(t)) \right) \textrm{d}t + G^h(m^h(t)) \,\textrm{d}W(t), \quad t \in [0,T], \end{aligned} \end{aligned}$$

(11)

with initial condition $m^h(0) = m_0^h:=m_0 \vert _{{\mathbb {Z}}_h} \in {\mathbb {E}}_h$, where $\sup _{h>0} |m_0^h|_{{\mathbb {E}}_h}$ is bounded under Assumption 2.1. In (10) and (11), $\kappa ^2, v(t)$ and W(t) are similarly the restrictions of the corresponding functions to ${\mathbb {Z}}_h$ for every $t \in [0,T]$. The term $S^h(m^h)$ is a discretised and a modified version of the Stratonovich correction S(m) . It is chosen to simplify the proof in Sect. 3.2 without affecting the limit. For example, $G_3^h(m^h)$ with the constant 2 does not match with the term $\gamma \langle m, \partial _x m \rangle m \times \partial _x m$ in ${\mathcal {G}}(u)$, but if $\langle m^h, \partial ^h m^{h-} \rangle$ converges to 0 in a suitable space, then the constant will be irrelevant to the final equation.

Remark 2.3

The semi-discrete scheme (11) is introduced in this paper as a tool for proving the existence, uniqueness and regularity of solutions. The proof can be easily adapted to the case of bounded interval with Neumann boundary conditions without requiring weighted Sobolev spaces in the convergence of approximations. A genuine numerical scheme for solving Eq. (8) will require discretisation in time such as using implicit Euler method or Crank Nicolson method. The fully discrete schemes need uniform bounds of time translate estimates to conclude convergence. Fully discrete schemes and their error estimates will be a subject of further research.

2.3 Main result

Definition 2.1

We say that a progressively measurable process m defined on $\left( \Omega ,{\mathcal {F}},\right. \left. \left( {\mathcal {F}}_t\right) ,{\mathbb {P}}, W\right)$ where W is a Wiener process, is a solution to Eq. (8) if

• (a) $|m(t,x)|=1$ (t, x)-a.e.

• (b) for every $T \in (0,\infty )$,

  $$\begin{aligned}{\mathbb {E}}\left[ \sup _{t\in [0,T]} \left| \partial _x m\right| ^2_{\mathbb L^2}+\int _0^T\left| \partial _{xx}m\right| ^2_{\mathbb L^2}\,\textrm{d}t \right] <\infty ,\end{aligned}$$

• (c) for every $t \in [0,T]$ the following equality holds in $\mathbb L^2$ :

  $$\begin{aligned} m(t)-m_0 = \int _0^t\left( F(m(s))+\frac{1}{2}S(m(s))\right) \,\textrm{d}s+\int _0^tG(m(s))\,\textrm{d}W(s),\quad {\mathbb {P}}\text {-a.s.} \nonumber \\ \end{aligned}$$

  (12)

Note that (a)–(d) above and Assumption 2.1 yield

$$\begin{aligned}\int _0^t\left( |F(m(s))|_{\mathbb L_2}^2+S(m(s))|_{\mathbb L^2}^2+|\kappa G(m(s))|^2_{\mathbb L^2}\right) \,\textrm{d}s<\infty ,\end{aligned}$$

hence the Bochner integral and the Itô integral in (12) are well defined in $\mathbb L^2$.

Lemma 2.4

There exists a unique solution $m^h$ of the semi-discrete scheme (11) in ${\mathcal {E}}_h$ satisfying $|m^h(t,x)|=1$, ${\mathbb {P}}$-a.s. for all $t \in [0,T]$ and $x \in {\mathbb {Z}}_h$.

Theorem 2.5

There exists a solution $(\Omega , {\mathcal {F}}, \left( {\mathcal {F}}_t\right) , {\mathbb {P}}, W, m)$ of (8) in the sense of Definition 2.1, such that for $p \in [1,\infty )$,

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t\in [0,T]} \left| \partial _x m(t)\right| ^p_{\mathbb L^2}+\left( \int _0^T\left| \partial _{xx}m(t)\right| ^2_{\mathbb L^2}\,\textrm{d}t\right) ^p \right] <\infty \,, \end{aligned}$$

provided that $C_\kappa$ is sufficiently small. Then for every $T> 0$ and $\alpha \in \left( 0,\frac{1}{2}\right)$,

$$\begin{aligned} m-m_0\in C^\alpha \left( [0,T];\,\mathbb L^2\right) , \quad {\mathbb {P}}\text {-a.s.} \end{aligned}$$

Moreover, there exists a convergent subsequence $\{m_h\}$ defined on $(\Omega , {\mathcal {F}}, {\mathbb {P}})$ such that $m_h$ has the same law as a quadratic interpolation of $m^h$ for every $h > 0$, and m is the ${\mathbb {P}}$-a.s. limit of $\{m_h\}$ in ${\mathcal {C}}([0,T]; {\mathbb {H}}^{-1}_{w})$ for some $w \ge 1$.

For the existence ($p=1$) and $L^p(\Omega )$-regularity in Theorem 2.5, it is sufficient to assume that

$$\begin{aligned} 4 C_\kappa \left( 1+\gamma ^2+ b_p^\frac{1}{p} (1+|\gamma |) \right) < \alpha \wedge 1, \end{aligned}$$

(13)

where $b_p$ is the constant in Burkholder–Davis–Gundy inequality. This gives an example of the required smallness of the noise, see (21) (or Remark 3.6) for a more precise condition (or bound) on $C_\kappa$.

Theorem 2.6

The solution m of (8) is pathwise unique and therefore unique in law.

3 Discretisation

From the definition of discrete operators and the discrete ${\mathbb {L}}_h^p$ norm, we deduce following results.

Remark 3.1

For $u: {\mathbb {Z}}_h \rightarrow {\mathbb {R}}^3$,

• (a) $\partial ^{h} (\partial ^{h} u) = \frac{1}{h} (\partial ^{h}u^+- \partial ^{h} u) = \Delta ^h u^+$,

• (b) for any $p\in [1,\infty ]$, $|u|_{{\mathbb {L}}_h^p} = |u^+|_{{\mathbb {L}}_h^p} = |u^-|_{{\mathbb {L}}_h^p}$, which implies $|\partial ^{h}u|_{{\mathbb {L}}_h^p} = |\partial ^{h} u^+|_{{\mathbb {L}}_h^p} = |\partial ^{h} u^-|_{{\mathbb {L}}_h^p}$, and hence,

  $$\begin{aligned} |\partial ^h u|_{\mathbb L_h^p}^2 \le \frac{4}{h^2} | u|_{\mathbb L_h^p}^2, \quad |\Delta ^h u|_{\mathbb L_h^p}^2 \le \frac{4}{h^2} |\partial ^hu|_{\mathbb L_h^p}^2, \end{aligned}$$

• (c) Lemma A.1 indicates $|u|_{\mathbb L_h^\infty }\le C |u|_{\mathbb H^1_h}$ for any $u\in \mathbb H_h^1 \,$.

3.1 Existence of a unique solution of the semi-discrete scheme

Lemma 3.2

For every $h >0$, if $f,g: {\mathbb {E}}_h \rightarrow {\mathbb {E}}_h$ are locally Lipschitz and satisfy $f(0) = g(0) = 0$, then $f \times g$, $\langle f, g \rangle$ and $\partial ^h f$ are also locally Lipschitz on ${\mathbb {E}}_h$.

The result in Lemma 3.2 is clear, and we omit the proof here. Then we check that the coefficients in (11) are locally Lipschitz on ${\mathbb {E}}_h$.

Lemma 3.3

For every $h > 0$, $F^h, G^h, S^h: {\mathbb {E}}_h \rightarrow {\mathbb {E}}_h$ are locally Lipschitz on ${\mathbb {E}}_h$.

Proof

Let $u,w \in {\mathbb {E}}_h$. It follows from Remark 3.1(b) that

$$\begin{aligned} |\partial ^h u - \partial ^h w|_{{\mathbb {E}}_h}^2&=|\partial ^h (u-w) |_{{\mathbb {L}}_h^\infty }^2 + |\partial ^{h} \partial ^h (u-w)|_{{\mathbb {L}}_h^2}^2\\&\le \frac{4}{h^2} \left( | u-w|_{{\mathbb {L}}_h^\infty }^2 + |\partial ^h (u-w)|_{{\mathbb {L}}_h^2}^2 \right) =\frac{4}{h^2} |u-w|_{{\mathbb {E}}_h}^2, \end{aligned}$$

and

$$\begin{aligned} |\Delta ^h u - \Delta ^h w|_{{\mathbb {E}}_h}^2&=|\Delta ^h (u-w) |_{{\mathbb {L}}_h^\infty }^2 + |\partial ^{h} \Delta ^h (u-w)|_{{\mathbb {L}}_h^2}^2\\&\le \frac{4}{h^2} \left( |\partial ^h (u-w)|_{{\mathbb {L}}_h^\infty }^2 + |\Delta ^h (u-w)|_{{\mathbb {L}}_h^2}^2 \right) \\&\le \frac{16}{h^4} \left( | u-w|_{{\mathbb {L}}_h^\infty }^2 + |\partial ^h (u-w)|_{{\mathbb {L}}_h^2}^2 \right) = \frac{16}{h^4} |u - w|_{{\mathbb {E}}_h}^2. \end{aligned}$$

By Lemma 3.2 and (7), $F^h, G^h$ and $S^h$ are locally Lipschitz. $\square$

Proof of Lemma 2.4

For each $n \in {\mathbb {N}}$ and $r^h =F^h, S^h$ and $G^h$, define

$$\begin{aligned} r_n^h(u) = {\left\{ \begin{array}{ll} r^h(u) & \text {if } |u|_{{\mathbb {E}}_h} \le n \\ r^h\left( \frac{n u}{|u|_{{\mathbb {E}}_h}} \right) & \text {if } |u|_{{\mathbb {E}}_h} > n. \end{array}\right. } \end{aligned}$$

Then $F_{n}^h$, $S_n^h$ and $G_n^h$ are Lipschitz on ${\mathbb {E}}_h$.

Fix $n \in {\mathbb {N}}$. Let $A_n: {\mathcal {E}}_h \rightarrow {\mathcal {E}}_h$ be given by

$$\begin{aligned} \begin{aligned} A_n(u)(t)&= m_0^h + \int _0^t \left( F_{n}^h(u(s)) + \frac{1}{2} S_n^h(u(s)) \right) \textrm{d}s + \int _0^t G_n^h (u(s)) \ \textrm{d}W(s) \\&= m_0^h + I_n(t) + J_n(t). \end{aligned} \end{aligned}$$

(14)

We first verify that $A_n(u) \in {\mathcal {E}}_h$ for $u \in {\mathcal {E}}_h$. Note that $F_n^h$ and $S_n^h$ are bounded on ${\mathbb {E}}_h$, with

$$\begin{aligned} {\mathbb {E}}\left[ |I_n(t)|_{{\mathbb {E}}_h}^2 \right]&\le T {\mathbb {E}}\left[ \int _0^t \left| F_{n}^h(u(s)) + \frac{1}{2} S_n^h(u(s)) \right| _{{\mathbb {E}}_h}^2 \textrm{d}s \right] \\&\le C_1(h,n) T {\mathbb {E}}\left[ \int _0^t |u(s)|_{{\mathbb {E}}_h}^2 \textrm{d}s \right] \\&\le C_1(h,n) T^2 |u|_{{\mathcal {E}}_h}^2, \end{aligned}$$

for some constant $C_1$ that depends on h and n. For $J_n(t)$, we have

$$\begin{aligned} \sum _j q_j^2 |f_j G^h(u(s))|_{{\mathbb {L}}_h^2}^2&= \sum _j q_j^2 \left| f_j u(s) \times \left( u(s) \times \partial ^h u(s) \right) + \gamma f_j u(s) \times \partial ^h u(s) \right| _{{\mathbb {L}}_h^2}^2 \\&\le 2 |\kappa ^2|_{{\mathbb {L}}_h^\infty } \left( |u(s)|_{{\mathbb {L}}_h^\infty }^4 + \gamma ^2 |u(s)|_{{\mathbb {L}}_h^\infty }^2 \right) |\partial ^h u(s)|^2_{{\mathbb {L}}_h^2}. \end{aligned}$$

where the last inequality holds by Tonelli’s theorem. This together with Remark 3.1(b) implies

$$\begin{aligned} \sum _j q_j^2 |\partial ^{h} (f_j G^h(u(s))) |_{{\mathbb {L}}_h^2}^2&\le \frac{4}{h^2} \sum _j q_j^2 |f_j G^h(u(s))|_{{\mathbb {L}}_h^2}^2 \\&\le \frac{8}{h^2} |\kappa ^2|_{{\mathbb {L}}_h^\infty } \left( |u(s)|_{{\mathbb {L}}_h^\infty }^4 + \gamma ^2 |u(s)|_{{\mathbb {L}}_h^\infty }^2 \right) \ |\partial ^{h} u(s)|^2_{{\mathbb {L}}_h^2}. \end{aligned}$$

Then by the definition of $G_n^h$, the assumption (6) and Fubini’s theorem,

$$\begin{aligned} {\mathbb {E}}\Bigg [ \int _0^t \sum _j q_j^2 \left| f_j G_n^h (u(s)) \right| ^2_{{\mathbb {H}}_h^1} \ \textrm{d}s \Bigg ]&\le 2 C_\kappa ^2 (n^4 + \gamma ^2 n^2) \left( 1+ \frac{4}{h^2} \right) T \sup _{s \in [0,t]} {\mathbb {E}}\left[ |u(s)|_{{\mathbb {E}}_h}^2 \right] \\&= C_2(h,n,\kappa ) T |u|_{{\mathcal {E}}_h}^2, \end{aligned}$$

for $C_2(h,n,\kappa ) = 2C_\kappa ^2 (n^4 + \gamma ^2 n^2) (1+\frac{4}{h^2}) T$. Thus, $J_n$ is a ${\mathbb {H}}_h^1$-valued continuous martingale. By Lemma A.1 (or Remark 3.1(c)), there exists a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} | J_n(t) |_{{\mathbb {E}}_h}^2&= | J_n(t) |_{{\mathbb {L}}_h^\infty }^2 + | \partial ^{h} J_n(t) |_{{\mathbb {L}}_h^2}^2 \le (C^2+1) |J_n(t) |_{{\mathbb {H}}_h^1}^2. \end{aligned} \end{aligned}$$

(15)

From [15, Corollary 4.29],

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\left[ \sup _{t \in [0,T]} \left| J_n(t) \right| _{{\mathbb {H}}_h^1}^2 \right]&= {\mathbb {E}}\left[ \sup _{t \in [0,T]} \left| \int _0^t G_n^h(u(s)) \ \textrm{d}W(s) \right| _{{\mathbb {H}}_h^1}^2 \right] \\&\le {\mathbb {E}}\left[ \int _0^T \sum _j q_j^2 \left| f_j G_n^h(u(s)) \right| _{{\mathbb {H}}_h^1}^2 \ \textrm{d}s \right] \\&\le C_2(h,n,\kappa ) T |u|_{{\mathcal {E}}_h}^2. \end{aligned} \end{aligned}$$

(16)

It follows from (15) and (16) that

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{t \in [0,T]} \left| J_n(t) \right| _{{\mathbb {E}}_h}^2 \right] \le \left( C^2+ 1\right) C_2(h,n,\kappa ) T |u|_{{\mathcal {E}}_h}^2. \end{aligned}$$

(17)

Thus, $A_n(u) \in {\mathcal {E}}_h$ for $u \in {\mathcal {E}}_h$.

For $u, \nu \in {\mathcal {E}}_h$, there exists a constant $C >0$ such that

$$\begin{aligned} |A_n(\nu ) - A_n(u)|_{{\mathcal {E}}_h}^2&\le C \sup _{t \in [0,T]} {\mathbb {E}} \left[ \left| \int _0^t F_{n}^h(\nu (s)) - F_{n}^h(u(s)) \ \textrm{d}s \right| _{{\mathbb {E}}_h}^2 \right] \\&\quad + C \sup _{t \in [0,T]} {\mathbb {E}} \left[ \left| \int _0^t \frac{1}{2} \left( S_n^h(\nu (s)) - S_n^h(u(s)) \right) \ \textrm{d}s \right| _{{\mathbb {E}}_h}^2 \right] \\&\quad + C \sup _{t \in [0,T]} {\mathbb {E}} \left[ \left| \int _0^t \left( G_n^h(\nu (s)) - G_n^h(u(s)) \right) \textrm{d}W(s) \right| _{{\mathbb {E}}_h}^2 \right] . \end{aligned}$$

Similarly, $M_n(t):= \int _0^t \left( G_n^h(\nu (s)) - G_n^h(u(s)) \right) \textrm{d}W(s)$ is a ${\mathbb {H}}_h^1$-valued continuous martingale, and replacing $J_n$ by $M_n$ in (15) and (16), we have

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{t \in [0,T]} \left| M_n(t) \right| _{{\mathbb {E}}_h}^2 \right] \le {\mathbb {E}} \left[ \int _0^T \sum _j q_j^2 \left| f_j \left( G_n^h(\nu (s)) - G_n^h(u(s))\right) \right| _{{\mathbb {H}}_h^1}^2 \ \textrm{d}s \right] . \nonumber \\ \end{aligned}$$

(18)

By construction, if $|\nu (s)|_{{\mathbb {E}}_h}, |u(s)|_{{\mathbb {E}}_h} \le n$, then

$$\begin{aligned} |G_n^h(\nu (s)) - G_n^h(u(s)) |_{{\mathbb {L}}_h^2}&\le |\nu (s) - u(s)|_{{\mathbb {L}}_h^\infty } \left( |\nu (s) |_{{\mathbb {L}}_h^\infty } + |u(s)|_{{\mathbb {L}}_h^\infty } +|\gamma | \right) |\partial ^h \nu (s)|_{{\mathbb {L}}_h^2} \\&\quad + \left( |u(s)|_{{\mathbb {L}}_h^\infty }^2 + |\gamma | |u(s)|_{{\mathbb {L}}_h^\infty } \right) | \partial ^h \nu (s) - \partial ^h u(s) |_{{\mathbb {L}}_h^2} \\&\le (3 n^2 + 2|\gamma |n) \ |\nu (s) - u(s) |_{{\mathbb {E}}_h}. \end{aligned}$$

If $|\nu (s)|_{{\mathbb {E}}_h}, |u(s)|_{{\mathbb {E}}_h} > n$, then let $n_s^\nu = n |\nu (s)|_{{\mathbb {E}}_h}^{-1}$ and $n_s^u = n |u(s)|_{{\mathbb {E}}_h}^{-1}$, we have

$$\begin{aligned} |n_s^\nu - n_s^u| \ |u|_{{\mathbb {E}}_h} \le |\nu (s)- u(s)|_{{\mathbb {E}}_h}, \end{aligned}$$

and

$$\begin{aligned}&|G_n^h(\nu (s)) - G_n^h(u(s)) |_{{\mathbb {L}}_h^2} \\&\quad = |G^h(n_s^\nu \nu (s)) - G^h(n_s^u u(s)) |_{{\mathbb {L}}_h^2} \\&\quad \le \left( n_s^\nu |\nu (s) - u(s)|_{{\mathbb {L}}_h^\infty } + | n_s^\nu - n_s^u| |u(s)|_{{\mathbb {L}}_h^\infty } \right) \\ &\qquad \left( n_s^\nu |\nu (s) |_{{\mathbb {L}}_h^\infty } + n_s^u |u(s)|_{{\mathbb {L}}_h^\infty } + |\gamma | \right) n_s^\nu |\partial ^h \nu (s)|_{{\mathbb {L}}_h^2} \\&\qquad + \left( n_s^\nu | \partial ^h \nu (s) - \partial ^h u(s) |_{{\mathbb {L}}_h^2} + | n_s^\nu - n_s^u| |\partial ^h u(s)|_{{\mathbb {L}}_h^2} \right) \left( (n_s^u)^2 |u(s)|_{{\mathbb {L}}_h^\infty }^2 + |\gamma | n_s^u |u(s)|_{{\mathbb {L}}_h^\infty } \right) \\&\quad \le 2 (3n^2 + 2|\gamma |n) \ |\nu (s) - u(s) |_{{\mathbb {E}}_h}. \end{aligned}$$

If $|\nu (s)|_{{\mathbb {E}}_h} > n$ and $|u(s)|_{{\mathbb {E}}_h} \le n$, then

$$\begin{aligned} |n_s^\nu -1 | \le n^{-1} |n-|\nu (s)|_{{\mathbb {E}}_h}| \le n^{-1} | |u(s)|_{{\mathbb {E}}_h} - |\nu (s)|_{{\mathbb {E}}_h} | \le n^{-1} |u(s)-\nu (s)|_{{\mathbb {E}}_h}, \end{aligned}$$

implying that $|n_s^\nu - 1| \ |u|_{{\mathbb {E}}_h} \le |u(s)- \nu (s)|_{{\mathbb {E}}_h}$ and

$$\begin{aligned} |G_n^h(\nu (s)) - G_n^h(u(s)) |_{{\mathbb {L}}_h^2}&\le 2 (3n^2 + 2|\gamma |n) \ |\nu (s) - u(s) |_{{\mathbb {E}}_h}. \end{aligned}$$

Similar result follows for $\partial ^h (G_n^h(\nu )-G_n^h(u))$ using Remark 3.1(b). Then, by (18), Lemma 3.3 and Hölder’s inequality, there exist constants $L_1(h,n,T)$ and $L_2(h,n,T)$ such that

$$\begin{aligned} |A_n(\nu ) - A_n(u)|_{{\mathcal {E}}_h}^2&\le L_1(h,n,T) \sup _{t \in [0,T]} {\mathbb {E}} \left[ \int _0^t |\nu (s) - u(s)|_{{\mathbb {E}}_h}^2 \ \textrm{d}s\right] \\&\le L_2(h,n,T) |\nu -u|_{{\mathcal {E}}_h}^2. \end{aligned}$$

Consider the discrete equation

$$\begin{aligned} \textrm{d}m^h_n(t) = \left( F_{n}^h(m^h_n(t)) + \frac{1}{2} S_n^h(m^h_n(t)) \right) \textrm{d}t + G_n^h(m^h_n(t)) \ \textrm{d}W(t), \end{aligned}$$

(19)

with $m^h_n(0) = m_0^h \in {\mathbb {E}}_h$, on intervals $[(k-1) {\tilde{T}},k{\tilde{T}}]$ for $k \ge 1$, where ${\tilde{T}}$ satisfies $L_2(h,n,{\tilde{T}}) < 1$. By the Banach fixed point theorem, there exists a unique solution $m_n^h \in {\mathcal {E}}_h$ of (19) on [0, T] .

Define the stopping times

$$\begin{aligned} \tau _n := \inf \{ t \ge 0 : |m_n^h(t)|_{{\mathbb {E}}_h}> n \}, \quad \tau _n' := \inf \{ t\ge 0 : |m_{n+1}^h(t)|_{{\mathbb {E}}_h} > n \}. \end{aligned}$$

Let $\tau = \tau _n \wedge \tau _n'$. Then $A_n(m_{n+1}^h) = A_{n+1}(m_{n+1}^h)$ on $[0,\tau )$, and by (18),

$$\begin{aligned}&{\mathbb {E}} \left[ \sup _{t \in [0,T]} \left| m_{n+1}^h(t \wedge \tau ) - m_n^h(t \wedge \tau ) \right| _{{\mathbb {E}}_h}^2 \right] \\&\quad = {\mathbb {E}} \left[ \sup _{t \in [0,T]} \left| A_n(m_{n+1}^h)(t \wedge \tau ) - A_n(m_n^h)(t \wedge \tau ) \right| _{{\mathbb {E}}_h}^2 \right] \\&\quad \le L_1(h,n,T) {\mathbb {E}} \left[ \int _0^T \left| m_{n+1}^h(s) - m_n^h(s) \right| _{{\mathbb {E}}_h}^2 \ \textrm{d}s \right] , \end{aligned}$$

which implies

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{t \in [0,T]} \left| m_{n+1}^h(t \wedge \tau ) - m_n^h(t \wedge \tau ) \right| _{{\mathbb {E}}_h}^2 \right] = 0, \end{aligned}$$

by Grönwall’s lemma. Thus, $m_{n+1}^h(\cdot \wedge \tau ) = m_n^h(\cdot \wedge \tau )$ and $\tau = \tau _n$, ${\mathbb {P}}$-a.s. and the discrete equation (11) admits a local solution $m^h(t) = m_n^h(t)$ for $t \in [0,\tau _n]$.

Recall that $m_0^h \in {\mathbb {E}}_h$ and $|m_0^h(x)| =1$ for all $x \in {\mathbb {Z}}_h$. Applying Itô’s lemma to $\frac{1}{2}|m^h(t,x)|^2$,

$$\begin{aligned} \frac{1}{2} d|m^h(t,x)|^2&= \left\langle F^h(m^h(t))(x) + \frac{1}{2} S^h(m^h(t))(x), m^h(t,x) \right\rangle \textrm{d}t \\&\quad + \frac{1}{2} \kappa ^2(x) |G^h(m^h(t))(x) |^2 \textrm{d}t \\&\quad + \left\langle G^h(m^h(t))(x), m^h(t,x) \right\rangle \textrm{d}W(t). \end{aligned}$$

By $\langle a, a \times b \rangle =0$, for any $t \in [0,\tau _n]$ and $x \in {\mathbb {Z}}_h$,

$$\begin{aligned}&\left\langle F^h(m^h), m^h \right\rangle (t,x) = 0, \\&\left\langle S^h(m^h), m^h \right\rangle (t,x) = \left\langle \kappa ^2 G^h_2(m^h), m^h \right\rangle (t,x) = - \kappa ^2(x) |G^h(m^h)|^2(t,x), \\&\left\langle G^h(m^h), m^h \right\rangle (t,x) = 0. \end{aligned}$$

Therefore, $|m^h(t,x)| = |m_n^h(t,x)| = |m_0^h(x)| = 1$ for any $t \in [0,\tau _n]$ and $x \in {\mathbb {Z}}_h$.

For any fixed h and n, the unique solution $m_n^h$ of (19) satisfies

$$\begin{aligned} {\mathbb {E}} \left[ |m^h_n(t \wedge \tau _n)|^2_{{\mathbb {E}}_h} \right]&= {\mathbb {E}} \left[ |m^h_n(t \wedge \tau _n)|_{{\mathbb {L}}_h^\infty }^2 + |\partial ^{h} m^h_n(t \wedge \tau _n) |_{{\mathbb {L}}_h^2}^2 \right] \\&= 1 + {\mathbb {E}} \left[ |\partial ^{h} m^h_n(t \wedge \tau _n) |_{{\mathbb {L}}_h^2}^2 \right] . \end{aligned}$$

We apply Itô’s lemma to $\frac{1}{2}|\partial ^{h} m_n^h(t \wedge \tau _n)|^2_{{\mathbb {L}}_h^2}$,

$$\begin{aligned}&\frac{1}{2}|\partial ^{h} m_n^h(t \wedge \tau _n)|^2_{{\mathbb {L}}_h^2} - \frac{1}{2}|\partial ^{h} m_0^h|^2_{{\mathbb {L}}_h^2} \\&\quad = \int _0^{t \wedge \tau _n} \left\langle -\Delta ^h m_n^h(s), F^h(m_n^h(s)) + \frac{1}{2} S^h(m_n^h(s)) \right\rangle _{{\mathbb {L}}_h^2} \textrm{d}s \\&\qquad + \int _0^{t \wedge \tau _n} \frac{1}{2} \sum _j q_j^2 \left| \partial ^{h}\left( f_j G^h(m_n^h(s))\right) \right| _{{\mathbb {L}}_h^2}^2 \textrm{d}s \\&\qquad + \int _0^{t \wedge \tau _n} \left\langle -\Delta ^h m_n^h(s), G^h(m_n^h(s)) \ \textrm{d}W(s) \right\rangle _{{\mathbb {L}}_h^2} . \end{aligned}$$

Since $|m_n^h(t,x)|=1$ for $(t,x) \in [0,\tau _n] \times {\mathbb {Z}}_h$, and

$$\begin{aligned} |\Delta ^h m_n^h(t)|_{{\mathbb {L}}_h^2}^2 + |\partial ^{h} m_n^h(t)|_{{\mathbb {L}}_h^2}^4 \le \frac{8}{h^2} |\partial ^{h} m_n^h(t)|_{{\mathbb {L}}_h^2}^2, \end{aligned}$$

there exist constants $\beta _1$ and $\beta _2$ that depend on h (not n) such that

$$\begin{aligned}&\left\langle -\Delta ^h m_n^h(s), F^h(m_n^h(s)) + \frac{1}{2} S^h(m_n^h(s)) \right\rangle _{{\mathbb {L}}_h^2} + \frac{1}{2} \sum _j q_j^2 \left| \partial ^{h}\left( f_j G^h(m_n^h(s)) \right) \right| _{{\mathbb {L}}_h^2}^2 \\&\quad \le \beta _1(h) |\partial ^{h} m_n^h(s)|_{{\mathbb {L}}_h^2}^2, \end{aligned}$$

and

$$\begin{aligned} \left\langle \Delta ^h m_n^h(s), G^h(m_n^h(s)) \right\rangle _{{\mathbb {L}}_h^2}^2&\le \frac{1}{4}\left( |m_n^h(s) \times \Delta ^h m_n^h(s)|^2_{{\mathbb {L}}_h^2} + |\partial ^h m_n^h(s)|^2_{{\mathbb {L}}_h^2} \right) ^2 \\&\le \beta _2(h) |\partial ^{h} m_n^h(s)|_{{\mathbb {L}}_h^2}^2. \end{aligned}$$

Then, by the boundedness of $\kappa ^2$, the stochastic integral $\int _0^{t \wedge \tau _n} \langle \Delta ^h m_n^h(s), G^h(m_n^h(s)) \textrm{d}W(s) \rangle _{{\mathbb {L}}_h^2}$ is a square integrable continuous martingale for $m_n^h \in {\mathcal {E}}_h$, for every $h>0$. Now, we have

$$\begin{aligned} |\partial ^{h} m_n^h(t \wedge \tau _n)|^2_{{\mathbb {L}}_h^2}&\le |\partial ^{h} m_0^h|^2_{{\mathbb {L}}_h^2} + 2\int _0^{t \wedge \tau _n} \beta _1(h) |\partial ^{h} m_n^h(s)|^2_{{\mathbb {L}}_h^2} \ \textrm{d}s \\&\quad - 2\int _0^{t \wedge \tau _n} \left\langle \Delta ^h m_n^h(s), G^h(m^h_n)(s) \ \textrm{d}W(s) \right\rangle _{{\mathbb {L}}_h^2} , \end{aligned}$$

where the stochastic integral part vanishes after taking expectation. We obtain

$$\begin{aligned} {\mathbb {E}} \left[ |m^h_n(t \wedge \tau _n)|^2_{{\mathbb {E}}_h} \right]&\le {\mathbb {E}} \left[ |m_0^h|^2_{{\mathbb {E}}_h} + 2\beta _1(h) \int _0^{t } |m^h_n(s\wedge \tau _n)|^2_{{\mathbb {E}}_h} \ \textrm{d}s \right] . \end{aligned}$$

By Grönwall’s lemma,

$$\begin{aligned} {\mathbb {E}} \left[ |m^h_n(t \wedge \tau _n)|^2_{{\mathbb {E}}_h} \right] \le {\mathbb {E}} \left[ |m_0^h|^2_{{\mathbb {E}}_h} \right] \exp \left( \int _0^{t} 2\beta _1(h) \textrm{d}s \right) \le K(h,t). \end{aligned}$$

(20)

By the definition of $\tau _n$, the left-hand side of (20) is greater than $n^2 {\mathbb {P}}(\tau _n \in [0,t])$, thus

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathbb {P}}(\tau _n \in [0,t]) \le \lim _{n \rightarrow \infty } K(h,t) n^{-2} = 0. \end{aligned}$$

In other words, $\tau _n \rightarrow \infty$, ${\mathbb {P}}$-a.s, as $n \rightarrow \infty$. Thus, the process $m^h(t) = \lim _{n \rightarrow \infty } m_n^h(t \wedge \tau _n)$ is the unique solution of the semi-discrete scheme (11). $\square$

3.2 Uniform estimates for the solution $m^h$ of the discrete SDE

For every $h>0$, let

$$\begin{aligned} M^h(t) := \int _0^t \langle \Delta ^h m^h(s), G^h(m^h(s))\ \textrm{d}W(s) \rangle _{{\mathbb {L}}_h^2}. \end{aligned}$$

In the following lemma, we deduce an upper bound of the stochastic integral $M^h(t)$ which is used in the proof of Lemma 3.5 to obtain uniform estimates for $m^h$.

Lemma 3.4

For any $p \in (0,\infty )$, there exists a constant $b_p$ independent of h such that

$$\begin{aligned} {\mathbb {E}} \left[ \sup _{t \in [0,T]} \left| M^h(t) \right| ^p \right]&\le \frac{1}{2} b_p (1+|\gamma |)^p C_{\kappa }^p \ \\&{\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^{2p} + \left( \int _0^T |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}t \right) ^p \right] . \end{aligned}$$

Proof

We observe that for every $j \ge 1$,

$$\begin{aligned}&\langle \Delta ^h m^h(t), q_j f_j G^h(m^h(t)) \rangle _{{\mathbb {L}}_h^2} \\&\quad = h \sum _x q_j f_j \langle m^h \times \Delta ^h m^h, m^h \times \partial ^h m^h + \gamma \partial ^h m^h \rangle (t,x)\\&\quad \le (1+|\gamma |) h \sum _x \left| q_j f_j \partial ^h m^h \right| \left| m^h \times \Delta ^h m^h \right| (t,x) \\&\quad \le (1+|\gamma |) \left( h \sum _x q_j^2 f_j^2 |\partial ^h m^h|^2(t,x) \right) ^{\frac{1}{2}} \left( h \sum _x |m^h \times \Delta ^h m^h|^2(t,x) \right) ^\frac{1}{2}, \end{aligned}$$

which implies

$$\begin{aligned} \sum _j \langle \Delta ^h m^h(t), q_j f_j G^h(m^h(t)) \rangle _{{\mathbb {L}}_h^2}^2&\le (1+|\gamma |)^2 \ |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ \\&\quad \left( h \sum _x \sum _j q_j^2 f_j^2 |\partial ^h m^h|^2(t,x) \right) \\&\le (1+|\gamma |)^2 \ C_\kappa ^2 \ |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^2. \end{aligned}$$

Then as in the proof of Lemma 2.4, for every fixed h,

$$\begin{aligned} \sum _j \langle \Delta ^h m^h(t), q_j f_j G^h(m^h(t)) \rangle _{{\mathbb {L}}_h^2}^2 < \infty ,\quad {\mathbb {P}} \text {-a.s.} \end{aligned}$$

implying that $M^h(t)$ is a continuous martingale. By the Burkholder–Davis–Gundy inequality, for $p \in (0,\infty )$, there exists a constant $b_p$ such that

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t \in [0,T]} | M^h(t) |^{p} \right]&\le b_{p} {\mathbb {E}}\left[ \left( \int _0^T \sum _j \left\langle \Delta ^h m^h(t), q_j f_j G^h(m^h(t)) \right\rangle _{{\mathbb {L}}_h^2}^2 \ \textrm{d}t \right) ^{\frac{p}{2}} \right] \\&\le b_{p} (1+|\gamma |)^p C_{\kappa }^{p} \ {\mathbb {E}}\\&\quad \left[ \left( \int _0^T \ |\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^2 \ |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}t \right) ^{\frac{p}{2}} \right] . \end{aligned}$$

Taking the supremum over t for $|\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^2$,

$$\begin{aligned}&{\mathbb {E}}\left[ \sup _{t \in [0,T]} | M^h(t) |^{p} \right] \\&\quad \le b_p (1+|\gamma |)^p C_{\kappa }^{p} \ {\mathbb {E}}\left[ \ \sup _{t \in [0,T]} |\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^p \ \left( \int _0^T |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}t \right) ^{\frac{p}{2}} \right] \\&\quad \le \frac{1}{2} b_p (1+|\gamma |)^p C_{\kappa }^{p} \ {\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^{2p} + \left( \int _0^T |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}t \right) ^p \right] . \end{aligned}$$

$\square$

Lemma 3.5

For any $p \in [1,\infty )$, assume that

$\{(q_j,f_j)\}_{j\ge 1}$ satisfies

$$\begin{aligned} \begin{aligned} N_{1,p}&:= 1 - 4^{p-1} b_p (1+|\gamma |)^p C_{\kappa }^{p}> 0,\\ N_{2,p}&:= 2^p \left( \alpha - (1+2\gamma ^2) C_\kappa ^2 - \delta \right) ^p - 4^{p-1} b_p (1+|\gamma |)^p C_{\kappa }^p > 0, \end{aligned} \end{aligned}$$

(21)

for some small $\delta > 0$, where $b_p$ is the constant in Lemma 3.4. Let $\sup _{h>0}|m_0^h|_{{\mathbb {E}}_h} \le K_0$. Then, there exist constants $K_{1,p}$ and $K_{2,p}$ that are independent of h, such that

$$\begin{aligned} & {\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \right] \le K_{1,p}, \end{aligned}$$

(22)

$$\begin{aligned} & \quad {\mathbb {E}}\left[ \left( \int _0^T |m^h(s) \times \Delta ^h m^h(s) |_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \right) ^p \right] \le K_{2,p}, \end{aligned}$$

(23)

for all $h> 0$.

Proof

As in Lemma 2.4, let $\phi (u) = \frac{1}{2} | \partial ^{h} u|_{{\mathbb {L}}_h^2}^2$ for $u \in {\mathbb {E}}_h$. Then, for $\nu ,w \in \mathring{{\mathbb {H}}}_h^1$,

$$\begin{aligned} \phi '(u) \nu&= \langle \partial ^{h} u, \partial ^{h} \nu \rangle _{{\mathbb {L}}_h^2} = - \langle \Delta ^h u, \nu \rangle _{{\mathbb {L}}_h^2}, \quad \phi ''(u)(\nu ,w) = \langle \partial ^{h}\nu , \partial ^{h} w \rangle _{{\mathbb {L}}_h^2}. \end{aligned}$$

By Itô’s lemma,

$$\begin{aligned} \begin{aligned} \frac{1}{2} | \partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^2- \frac{1}{2} | \partial ^{h} m^h(0)|_{{\mathbb {L}}_h^2}^2&=\phi (m^h(t)) - \phi (m^h(0)) \\&= - \int _0^t \left\langle \Delta ^h m^h(s), F^h(m^h)(s) \right\rangle _{{\mathbb {L}}_h^2} \textrm{d}s \\&\quad - \int _0^t \left\langle \Delta ^h m^h(s), \frac{1}{2} S^h(m^h)(s)\right\rangle _{{\mathbb {L}}_h^2} \textrm{d}s \\&\quad + \frac{1}{2} \int _0^t \sum _j | \partial ^{h} (q_j f_j G^h(m^h)(s) ) |_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \\&\quad - \int _0^t \langle \Delta ^h m^h(s), G^h(m^h)(s) \ \textrm{d}W(s) \rangle _{{\mathbb {L}}_h^2} \\&:= \int _0^t \left( T_1(s) + T_2(s) + T_3(s)\right) \,\textrm{d}s - M^h(t), \end{aligned} \end{aligned}$$

(24)

where $M^h(t)$ is already estimated in Lemma 3.4.

$\underline{\text { An estimate on }\,\,T_1:}$

$$\begin{aligned} \begin{aligned} T_1(s)&= -\langle \Delta ^h m^h(s), F^h(m^h)(s) \rangle _{{\mathbb {L}}_h^2} \\&= \alpha \left\langle \Delta ^h m^h(s), m^h(s) \times \left( m^h(s) \times \Delta ^h m^h(s)\right) \right\rangle _{{\mathbb {L}}_h^2} \\&\quad -\left\langle \Delta ^h m^h(s), m^h(s) \times \left( m^h(s) \times v^h(s)\partial ^{h} m^h(s) + \gamma v^h(s)\partial ^{h} m^h(s)\right) \right\rangle _{{\mathbb {L}}_h^2} \\&= - \alpha |m^h(s) \times \Delta ^h m^h(s) |^2_{{\mathbb {L}}_h^2} \\&\quad + \left\langle m^h(s) \times \Delta ^h m^h(s), m^h(s) \times v^h(s)\partial ^{h} m^h(s) + \gamma v^h(s)\partial ^{h} m^h(s) \right\rangle _{{\mathbb {L}}_h^2}. \end{aligned} \end{aligned}$$

(25)

The second term on the right-hand side of (25) is estimated using (7) and the fact that $|m^h|=1$ ${\mathbb {P}}$-a.s., as follows

$$\begin{aligned} \begin{aligned}&\left\langle m^h(s) \times \Delta ^h m^h(s), m^h(s) \times v^h(s)\partial ^{h} m^h(s) + \gamma v^h(s)\partial ^{h} m^h(s) \right\rangle _{{\mathbb {L}}_h^2} \\&\quad \le \varepsilon ^2 |m^h(s) \times \Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2 + \frac{1}{2\varepsilon ^2} C_v^2 (1+\gamma ^2) |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^2}^2, \end{aligned} \end{aligned}$$

(26)

for arbitrary $\varepsilon >0$. An estimate of $T_1$ is obtained from (25) and (26)

$$\begin{aligned} T_1 \le (\varepsilon ^2 - \alpha )|m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 + \frac{1}{2\varepsilon ^2} C_v^2 (1+\gamma ^2) |\partial ^{h} m^h(t)|_{{\mathbb {L}}_h^2}^2. \end{aligned}$$

(27)

$\underline{\text { An estimate on }\,\,T_2:}$

$$\begin{aligned} T_2(s)&=-\frac{1}{2}\langle \Delta ^h m^h(s), S^h(m^h(s)) \rangle _{{\mathbb {L}}_h^2} \\&= \frac{1}{2}\left\langle m^h\times \Delta ^h m^h, \kappa \kappa ' \left( m^h\times G^h(m^h)+ \gamma G^h(m^h)\right) \right\rangle _{{\mathbb {L}}_h^2} \\&\quad - \frac{1}{2} \left\langle \Delta ^h m^h, {\mathcal {G}}^h_\kappa (m^h) \right\rangle _{{\mathbb {L}}_h^2} \\&= T_{21} + T_{22}. \end{aligned}$$

Using $|m^h|=1$, ${\mathbb {P}}$-a.s., we estimate $T_{21}$:

$$\begin{aligned} T_{21}&= \frac{1}{2} \left\langle m^h \times \Delta ^h m^h, \kappa \kappa ' \left[ (\gamma ^2-1) m^h \times \partial ^h m^h + 2 \gamma m^h \times (m^h \times \partial ^h m^h) \right] \right\rangle _{{\mathbb {L}}_h^2}\nonumber \\&\le \frac{1}{2} \varepsilon ^2 |m^h(s)\times \Delta ^h m^h(s)|^2_{{\mathbb {L}}_h^2} + \frac{1}{4\varepsilon ^2}((\gamma ^2-1)^2+ 4\gamma ^2) \ |\kappa \kappa '|_{{\mathbb {L}}_h^\infty }^2 |\partial ^{h} m^h(s)|^2_{{\mathbb {L}}_h^2}. \end{aligned}$$

(28)

where $|\kappa \kappa '|_{{\mathbb {L}}_h^\infty }^2 \le C_\kappa ^4$ by (6).

To estimate $T_{22}$, we first note that for any $u:{\mathbb {Z}}_h \rightarrow {\mathbb S}^2$,

$$\begin{aligned} {\mathcal {G}}^h_\kappa (u)&= \frac{1}{2}\left( (\kappa ^2)^- + \kappa ^2 \right) \left[ (\gamma ^2 -1) u \times \left( u \times \Delta ^h u\right) - 2 \gamma u \times \Delta ^h u \right] \\&\quad - \gamma ^2 \kappa ^2 \partial ^h u \times (u \times \partial ^h u) \\&\quad - \kappa ^2 |u \times \partial ^h u|^2 u + 2\gamma (\kappa ^2)^- \langle u, (\partial ^h u)^- \rangle u \times \partial ^h u. \end{aligned}$$

By (98), we deduce

$$\begin{aligned} T_{22}(s)&= - \frac{1}{2}h \sum _x \left\langle \Delta ^h m^h, {\mathcal {G}}^h_\kappa (m^h)\right\rangle (s,x)\nonumber \\&= \frac{1}{4}(\gamma ^2-1) h \sum _x \left( (\kappa ^2)^- + \kappa ^2 \right) |m^h\times \Delta ^h m^h|^2 (s,x)\nonumber \\&\quad + \frac{1}{2}\gamma ^2 h \sum _x \kappa ^2 \left\langle \Delta m^h, \partial ^h m^h \times (m^h \times \partial ^h m^h) \right\rangle (s,x) \nonumber \\&\quad - \frac{1}{4} h \sum _x \kappa ^2 |m^h \times \partial ^h m^h|^2 \left( |\partial ^{h} m^h|^2 + |(\partial ^{h} m^h)^-|^2\right) (s,x)\nonumber \\&\quad - \gamma h \sum _x (\kappa ^2)^- \left\langle \Delta ^h m^h, m^h \times \partial ^h m^h \right\rangle \left\langle m^h, (\partial ^h m^h)^- \right\rangle (s,x) \nonumber \\&= T_{22a}(s) + T_{22b}(s) + T_{22c}(s) + T_{22d}(s). \end{aligned}$$

(29)

It is clear that

$$\begin{aligned} T_{22a}(s)&= \frac{1}{4}(\gamma ^2-1) h \sum _x \left( (\kappa ^2)^- + \kappa ^2 \right) |m^h\times \Delta ^h m^h|^2 (s,x) \nonumber \\&\le \frac{1}{2} \gamma ^2 |\kappa |^2_{{\mathbb {L}}_h^\infty } |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2}(s) - \frac{1}{4} h \sum _x \left( (\kappa ^2)^- + \kappa ^2 \right) |m^h\times \Delta ^h m^h|^2 (s,x), \end{aligned}$$

(30)

where the second term on the right-hand side will cancel with parts of $T_3$.

To estimate $T_{22b}$, we observe that for any $u: {\mathbb {Z}}_h \rightarrow {\mathbb S}^2$, using (98),

$$\begin{aligned} \left\langle \Delta ^h u, \partial ^h u \times (u \times \partial ^h u) \right\rangle&= |\partial ^h u|^2 \left\langle u, \Delta ^h u \right\rangle - \left\langle u, \partial ^h u \right\rangle \left\langle \partial ^h u, \Delta ^h u \right\rangle \nonumber \\&= -\frac{1}{2} |\partial ^h u|^2 \left( |\partial ^h u|^2 + |\partial ^h u^-|^2 \right) \nonumber \\&\quad + \frac{1}{2} |\partial ^h u|^2 \left( |\partial ^h u|^2 - \left\langle \partial ^h u, \partial ^h u^- \right\rangle \right) \nonumber \\&= -\frac{1}{2} |\partial ^h u|^2 |\partial ^h u^-|^2 - \frac{1}{2} |\partial ^h u|^2 \left\langle \partial ^h u, \partial ^h u^- \right\rangle , \end{aligned}$$

(31)

where

$$\begin{aligned} \left\langle \partial ^h u, \partial ^h u^- \right\rangle = \frac{1}{2} \left( |\partial ^h u|^2 + |\partial ^h u^-|^2 -h^2 |\Delta ^h u|^2 \right) . \end{aligned}$$

(32)

If $|\partial ^h u(x)| \le |\partial ^h u^-(x)|$ at some $x \in {\mathbb {Z}}_h$, then

$$\begin{aligned} -\left\langle \partial ^h u, \partial ^h u^- \right\rangle (x) \le |\partial ^h u^-(x)|^2, \end{aligned}$$

and by (31),

$$\begin{aligned} \left\langle \Delta ^h u, \partial ^h u \times (u \times \partial ^h u) \right\rangle (x)&\le -\frac{1}{2} |\partial ^h u(x)|^2 |\partial ^h u^-(x)|^2 + \frac{1}{2} |\partial ^h u(x)|^2 |\partial ^h u^-(x)|^2 \\&= 0. \end{aligned}$$

If $|\partial ^h u(x)| \ge |\partial ^h u^-(x)|$, then we can show that the term given by (31) is bounded by $|u \times \Delta ^h u|^2(x)$. Explicitly, by (32),

$$\begin{aligned}&|u \times \Delta ^h u|^2(x) - \left\langle \Delta ^h u, \partial ^h u \times (u \times \partial ^h u) \right\rangle (x)\\&\quad = |\Delta ^h u|^2 - \left\langle u, \Delta ^h u \right\rangle ^2 + \frac{1}{2} |\partial ^h u|^2 |\partial ^h u^-|^2 + \frac{1}{2} |\partial ^h u|^2 \left\langle \partial ^h u, \partial ^h u^- \right\rangle \\&\quad = |\Delta ^h u|^2 - \frac{1}{4} \left( |\partial ^h u|^2 + |\partial ^h u^-|^2 \right) ^2 \\&\qquad + \frac{1}{2} |\partial ^h u|^2 |\partial ^h u^-|^2 + \frac{1}{4} |\partial ^h u|^2 \left( |\partial ^h u|^2 + |\partial ^h u^-|^2 -h^2 |\Delta ^h u|^2 \right) \\&\quad = \left( 1- \frac{1}{4}h^2 |\partial ^h u|^2 \right) |\Delta ^h u|^2 - \frac{1}{4} |\partial ^h u^-|^4 + \frac{1}{4} |\partial ^h u|^2 |\partial ^h u^-|^2 \\&\quad \ge 0, \end{aligned}$$

where the last inequality holds by $h^2 |\partial ^h u|^2 \le 4$ and $|\partial ^h u(x)| \ge |\partial ^h u^-(x)|$. Combining the two cases and replacing u by $m^h(s)$, we have

$$\begin{aligned} T_{22b}(s)&= \frac{1}{2} \gamma ^2 h \sum _x \kappa ^2 \left\langle \Delta ^h m^h, \partial ^h m^h \times (m^h \times \partial ^h m^h) \right\rangle (s,x) \nonumber \\&\le \frac{1}{2} \gamma ^2 h \sum _x \kappa ^2 |m^h \times \Delta ^h m^h|^2(s,x) \ \mathbbm {1}_{\{ |\partial ^h m^h(s,x)| \ge |\partial ^h {m^h}^-(s,x)| \}} \nonumber \\&\le \frac{1}{2} \gamma ^2 C_\kappa ^2 \ |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2}(s). \end{aligned}$$

(33)

We will see later in the proof that $T_{22c}$ and $T_{22d}$ also cancel with parts of $T_3$.

$\underline{\text { An estimate on } \,\,T_3:}$

$$\begin{aligned} T_3(s)&= \frac{1}{2} \sum _j \left| \partial ^{h} (q_jf_j G^h(m^h)) \right| ^2_{{\mathbb {L}}_h^2} (s) \\&= \frac{1}{2} \sum _j q_j^2 \left| \partial ^{h} (f_j G^h(m^h)) \right| ^2_{{\mathbb {L}}_h^2} (s) \\&= \frac{1}{2} h \sum _x \sum _j q_j^2 \left| \partial ^{h} \left( f_j m^h \times (m^h \times \partial ^{h} m^h) \right) \right| ^2(s,x) \\&\quad + \frac{1}{2} \gamma ^2 h \sum _x \sum _j q_j^2 \left| \partial ^{h} (f_j m^h \times \partial ^{h}m^h) \right| ^2(s,x) \\&\quad + \gamma h \sum _x \sum _j q_j^2 \left\langle \partial ^{h} \left( f_j m^h \times (m^h \times \partial ^{h} m^h) \right) , \partial ^{h} \left( f_j m^h \times \partial ^{h} m^h \right) \right\rangle (s,x) \\&= T_{31}(s) + T_{32}(s) + T_{33}(s). \end{aligned}$$

We first estimate $T_{31}(s)$. For $u: {\mathbb {Z}}_h \rightarrow {\mathbb S}^2$, we have for every $j \ge 1$,

$$\begin{aligned}&\partial ^{h} \left( f_j u \times (u \times \partial ^{h} u) \right) (x) \\&\quad = \frac{1}{2} \left( \partial ^{h} f_j \ u^+ \times (u^+ \times \partial ^{h} u^+) + f_j \partial ^{h} \left( u \times (u \times \partial ^{h} u) \right) \right) (x) \\&\qquad + \frac{1}{2} \left( \partial ^{h}f_j \ u \times (u \times \partial ^{h} u) + f_j^+ \partial ^{h} \left( u \times (u \times \partial ^{h} u) \right) \right) (x) \\&\quad = \frac{1}{2} \partial ^{h}f_j \left( u^+ \times (u^+ \times \partial ^{h} u^+) + u \times (u \times \partial ^{h} u) \right) (x) \\&\qquad + \frac{1}{2}\left[ f_j \partial ^{h}u \times (u \times \partial ^{h} u) + f_j^+ u \times \partial ^{h}(u \times \partial ^{h} u) \right] (x) \\&\qquad + \frac{1}{2}\left[ f_j^+ \partial ^{h}u \times (u^+ \times \partial ^{h} u^+) + f_j u^+ \times \partial ^{h}(u \times \partial ^{h} u) \right] (x) \\&\quad = \frac{1}{2}A_0(x) + \frac{1}{2} \left[ A_1(x) + B_1(x) \right] + \frac{1}{2} \left[ A_2(x) + B_2(x) \right] , \end{aligned}$$

where

$$\begin{aligned} A_0(x)&= \partial ^{h}f_j \left( u^+ \times (u^+ \times \partial ^{h} u^+) + u \times (u \times \partial ^{h} u) \right) (x), \\ A_1(x)&= f_j \partial ^{h}u \times (u \times \partial ^{h} u), \\ A_2(x)&= f_j^+ \partial ^{h}u \times (u^+ \times \partial ^{h} u^+), \\ B_1(x)&= f_j^+ u \times \partial ^{h}(u \times \partial ^{h} u), \\ B_2(x)&= f_j u^+ \times \partial ^{h}(u \times \partial ^{h} u). \end{aligned}$$

Hence,

$$\begin{aligned}&\frac{1}{2} \left| \partial ^{h} \left( f_j u \times (u \times \partial ^{h} u) \right) \right| ^2(x) \\&\quad = \frac{1}{2} \left( \frac{1}{4} |A_0|^2(x) + \frac{1}{4} |A_1 + B_1 + A_2 + B_2|^2(x) + \frac{1}{2} \left\langle A_0, A_1 + A_2 + B_1 + B_2 \right\rangle (x) \right) \\&\quad \le \frac{1}{2} \left( \frac{1}{4} |A_0|^2(x) + \frac{1}{2} |A_1 + B_1|^2(x) + \frac{1}{2} |A_2 + B_2|^2(x) + \frac{1}{2} \left\langle A_0, A_1 + A_2 + B_1 + B_2 \right\rangle (x) \right) \\&\quad = \frac{1}{8} |A_0|^2(x) + \frac{1}{4} \left( |A_1|^2 + |B_1|^2 + |A_2|^2 + |B_2|^2 \right) (x) \\&\qquad + \frac{1}{2}\left\langle A_1, B_1 \right\rangle (x) + \frac{1}{2} \left\langle A_2, B_2 \right\rangle (x) + \frac{1}{4} \left\langle A_0, A_1 + A_2 + B_1 +B_2 \right\rangle (x). \end{aligned}$$

For the square $\frac{1}{8} |A_0|^2(x)$:

$$\begin{aligned} \begin{aligned} \frac{1}{8} h \sum _x \sum _j q_j^2 |A_0|^2(x)&= \frac{1}{8} h \sum _x \sum _j q_j^2 | \partial ^{h} f_j |^2 \ \left| u^+ \times (u^+ \times \partial ^{h} u^+) + u \times (u \times \partial ^{h} u) \right| ^2(x) \\&\le \frac{1}{8} C_\kappa ^2 \ h \sum _x \left| u^+ \times (u^+ \times \partial ^{h} u^+) + u \times (u \times \partial ^{h} u) \right| ^2(x) \\&\le \frac{1}{2} C_\kappa ^2 \ |\partial ^{h} u|_{{\mathbb {L}}_h^2}^2, \end{aligned} \end{aligned}$$

(34)

where the second inequality holds by applying the Mean Value Theorem to $f_j$ on the interval $[x,x+h]$ for every $j \ge 1$, such that there exists some $\xi _h \in (x,x+h)$ satisfying

$$\begin{aligned} \left| \frac{f_j(x+h)-f_j(x)}{h} \right| = |f_j'(\xi _h)|, \end{aligned}$$

and $|\sum _j q_j^2 (f_j')^2|_{{\mathbb {L}}^\infty } \le C_\kappa ^2$ by assumption (6).

For the squares $\frac{1}{4}|A_1|^2(x)$ and $\frac{1}{4}|A_2|^2(x)$:

$$\begin{aligned} \begin{aligned}&\frac{1}{4} h \sum _x \sum _j q_j^2 \left( |A_1|^2(x) + |A_2|^2(x) \right) \\&\quad = \frac{1}{4} \sum _x \sum _j q_j^2 \left( \left| f_j \partial ^{h}u \times (u \times \partial ^{h} u) \right| ^2 + \left| f_j^+ \partial ^{h} u \times (u^+ \times \partial ^{h} u^+) \right| ^2 \right) (x) \\&\quad \le \frac{1}{4} h \sum _x \sum _j q_j^2 f_j^2 \ |\partial ^{h}u|^2 \ |u \times \partial ^{h}u|^2(x)\\&\qquad + \frac{1}{4} h \sum _x \sum _j q_j^2 (f_j^+)^2 \ |\partial ^{h}u|^2 \ |u^+ \times \partial ^{h} u^+|^2(x) \\&\quad = \frac{1}{4} h \sum _x \left( \sum _j q_j^2 f_j^2 \right) |u \times \partial ^{h} u|^2 \left( |\partial ^{h}u|^2 + |\partial ^h u^-|^2 \right) (x) \\&\quad = \frac{1}{4} h \sum _x \kappa ^2 |u \times \partial ^{h} u|^2 \left( |\partial ^{h}u|^2 + |\partial ^h u^-|^2 \right) (x), \end{aligned} \end{aligned}$$

(35)

where the right-hand side cancels with $T_{22c}(s)$ in (29) when u is replaced with $m^h(s)$.

For the squares $\frac{1}{4}|B_1|^2(x)$ and $\frac{1}{4}|B_2|^2(x)$, we first observe that

$$\begin{aligned} \begin{aligned} \partial ^{h} \left( u \times \partial ^{h}u\right) (x)&= u^+\times \Delta ^hu^+. \end{aligned} \end{aligned}$$

(36)

Then,

$$\begin{aligned}&\frac{1}{4}\left( |B_1|^2(x) + |B_2|^2(x) \right) \\&\quad = \frac{1}{4} (f_j^+)^2 \left| u \times \left( u^+ \times \Delta ^h u^+ \right) \right| ^2(x) + \frac{1}{4} f_j^2 \left| u^+ \times \left( u^+ \times \Delta ^h u^+ \right) \right| ^2(x) \\&\quad \le \frac{1}{4} \left( (f_j^+)^2 + f_j^2 \right) \left| u^+ \times \Delta ^h u^+ \right| ^2(x). \end{aligned}$$

This implies that

$$\begin{aligned} \begin{aligned} \frac{1}{4} h \sum _x \sum _j q_j^2 \left( |B_1|^2(x) + |B_2|^2(x) \right)&= \frac{1}{4} h \sum _x \sum _j q_j^2 \left( (f_j^-)^2 + f_j^2 \right) \left| u \times \Delta ^h u \right| ^2(x) \\&= \frac{1}{4} h \sum _x \left( (\kappa ^2)^- + \kappa ^2 \right) \left| u \times \Delta ^h u \right| ^2(x), \end{aligned} \end{aligned}$$

(37)

where the right-hand side cancels with a part of the estimate for $T_{22a}$ in (30) when $u = m^h(s)$ as aforementioned.

For the cross terms $\frac{1}{2} \left\langle A_1,B_1 \right\rangle (x)$ and $\frac{1}{2} \left\langle A_2,B_2 \right\rangle (x)$:

$$\begin{aligned} \left\langle A_1, B_1 \right\rangle (x)&= f_j f_j^+ \left\langle \partial ^{h} u \times (u \times \partial ^{h}u), u \times \partial ^{h}(u \times \partial ^{h} u) \right\rangle (x) \\&= f_j f_j^+ \left\langle |\partial ^{h} u|^2 u - \langle \partial ^{h}u, u \rangle \partial ^hu, u \times \partial ^{h}(u \times \partial ^{h} u) \right\rangle (x) \\&= f_j f_j^+ \langle u, \partial ^{h} u \rangle \left\langle u \times \partial ^{h}u, \partial ^{h}(u \times \partial ^{h}u) \right\rangle (x), \end{aligned}$$

and similarly,

$$\begin{aligned} \left\langle A_2, B_2 \right\rangle (x)&= f_j f_j^+ \left\langle \partial ^{h}u \times (u^+ \times \partial ^{h}u^+), u^+ \times \partial ^{h}(u \times \partial ^{h}u) \right\rangle (x) \\&= f_j f_j^+ \langle u^+, \partial ^{h}u \rangle \left\langle u^+ \times \partial ^{h}u^+, \partial ^{h}(u \times \partial ^{h}u) \right\rangle (x). \end{aligned}$$

Then,

$$\begin{aligned} \begin{aligned}&\left\langle A_1,B_1 \right\rangle + \left\langle A_2,B_2 \right\rangle (x) \\&\quad = f_j f_j^+ \left\langle \langle u, \partial ^{h} u \rangle u \times \partial ^{h}u + \langle u^+, \partial ^{h}u \rangle u^+ \times \partial ^{h}u^+, \ \partial ^{h}(u \times \partial ^{h}u) \right\rangle (x). \end{aligned} \end{aligned}$$

(38)

By (36), the left term in the inner product (38) can be simplified as

$$\begin{aligned}&\langle u, \partial ^{h} u \rangle u \times \partial ^{h}u + \langle u^+, \partial ^{h}u \rangle u^+ \times \partial ^{h}u^+ \\&\quad = \left( \langle u,\partial ^{h}u \rangle + \langle u^+, \partial ^{h}u \rangle \right) u \times \partial ^{h}u + \langle u^+, \partial ^{h} u \rangle h \left( u^+ \times \Delta ^hu^+ \right) \\&\quad = \langle u^+, u^+-u \rangle \left( u^+ \times \Delta ^hu^+ \right) , \end{aligned}$$

where the second equality holds by observing $\langle u+u^+, \partial ^{h}u \rangle (x) = 0$ due to $|u(x)| = 1$ for all x. Recall that the right term in the inner product (38) is $\partial ^{h}(u \times \partial ^{h}u) = u^+ \times \Delta ^hu^+$ by (36). Then,

$$\begin{aligned} \frac{1}{2} \left( \left\langle A_1,B_1 \right\rangle + \left\langle A_2,B_2 \right\rangle (x) \right)&= \frac{1}{2} f_j f_j^+ \langle u^+, u^+-u \rangle \left| u^+ \times \Delta ^hu^+ \right| ^2(x) \\&\le \frac{1}{2} \left( f_j^2 + (f_j^+)^2 \right) |u^+ \times \Delta ^hu^+|^2(x) \end{aligned}$$

Taking the sum over $x \in {\mathbb {Z}}_h$,

$$\begin{aligned} \begin{aligned} \frac{1}{2} h \sum _x \sum _j q_j^2 \left( \left\langle A_1, B_1 \right\rangle + \left\langle A_2, B_2 \right\rangle \right) (x)&\le \frac{1}{2} h \sum _x \left( \kappa ^2 + (\kappa ^2)^+ \right) |u^+ \times \Delta ^hu^+|^2(x) \\&\le C_\kappa ^2 \ |u \times \Delta ^hu|_{{\mathbb {L}}_h^2}^2. \end{aligned} \end{aligned}$$

(39)

For the cross term $\frac{1}{4}\left\langle A_0, \ A_1 + A_2 + B_1 + B_2\right\rangle (x)$:

$$\begin{aligned} A_0(x)&= \frac{f_j^+-f_j}{h} \left( u^+ \times (u^+ \times \partial ^{h}u^+) + u \times (u \times \partial ^{h} u) \right) (x), \end{aligned}$$

and

$$\begin{aligned} (A_1 + A_2 + B_1 + B_2)(x)&= (f_j+f_j^+) \partial ^{h}\left( u \times (u \times \partial ^{h}u) \right) (x) \\&= \frac{f_j+f_j^+}{h} \left( u^+ \times (u^+ \times \partial ^{h}u^+) - u \times (u \times \partial ^{h} u) \right) (x), \end{aligned}$$

which imply

$$\begin{aligned}&\frac{1}{4} \left\langle A_0, \ A_1 + A_2 + B_1 + B_2 \right\rangle (x) \\&\quad = \frac{1}{4h^2} \left( (f_j^+)^2 - f_j^2 \right) \left( |u^+ \times (u^+\times \partial ^{h}u^+)|^2 - |u \times (u \times \partial ^{h}u)|^2 \right) (x). \end{aligned}$$

Then, using again the Mean Value Theorem for $\Delta ^h (f_j)^2$,

$$\begin{aligned} \begin{aligned}&\frac{1}{4} h \sum _x \sum _j q_j^2 \left\langle A_0, \ A_1 + A_2 + B_1 + B_2 \right\rangle (x) \\&\quad = \frac{1}{4} h \sum _x \sum _j q_j^2 \frac{1}{h^2}\left( f_j^2 - (f_j^-)^2 - (f_j^+)^2 + f_j^2 \right) |u \times (u \times \partial ^{h} u)|^2(x) \\&\quad = -\frac{1}{4} h \sum _x \sum _j q_j^2 \Delta ^h \left( f_j^2 \right) |u \times (u \times \partial ^{h} u)|^2(x) \\&\quad \le \frac{1}{2}C_\kappa ^2 \ |\partial ^{h} u|_{{\mathbb {L}}_h^2}^2. \end{aligned} \end{aligned}$$

(40)

Therefore, by (34), (35), (37), (39) and (40),

$$\begin{aligned} \begin{aligned} T_{31}(s)&= \frac{1}{2} h \sum _x \left| \partial ^{h} \left( f_j m^h \times (m^h \times \partial ^{h}m^h) \right) \right| ^2(s,x) \\&\le C_\kappa ^2 |\partial ^{h} m^h|^2_{{\mathbb {L}}_h^2}(s) + C_\kappa ^2 |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2}(s) \\&\quad + \frac{1}{4} h \sum _x \kappa ^2 |m^h \times \partial ^{h}m^h|^2 \left( |\partial ^{h}m^h|^2 + |(\partial ^h m^h)^-|^2 \right) (s,x) \\&\quad + \frac{1}{4} h \sum _x \left( \kappa ^2 +(\kappa ^2)^- \right) |m^h \times \Delta ^h m^h|^2(s,x). \end{aligned} \end{aligned}$$

(41)

Next, we estimate $T_{32}(s)$. Using (36),

$$\begin{aligned} \begin{aligned} T_{32}(s)&= \frac{1}{2} \gamma ^2 h \sum _x \sum _j q_j^2 |\partial ^{h}(f_j m^h \times \partial ^{h}m^h)|^2(s,x) \\&= \frac{1}{2} \gamma ^2 h \sum _x \sum _j q_j^2 |(\partial ^{h} f_j) m^h \times \partial ^{h}m^h + f_j^+ (m^h \times \Delta ^h m^h)^+|^2(s,x) \\&\le \gamma ^2 h \sum _x \sum _j q_j^2 |\partial ^{h} f_j|^2 \ |\partial ^{h}m^h|^2(s,x) \\&\quad + \gamma ^2 h \sum _x \left( \sum _j q_j^2 f_j^2 \right) |m^h \times \Delta ^h m^h|^2(s,x) \\&\le \gamma ^2 C_\kappa ^2 \left( |\partial ^{h} m^h|^2_{{\mathbb {L}}_h^2}(s) + |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2} \right) . \end{aligned} \end{aligned}$$

(42)

Finally, we estimate $T_{33}(s)$. We note that for $u = u(x)$ with $|u(x)|=1$ for all x and for all $j \ge 1$,

$$\begin{aligned}&\left\langle \partial ^{h} \left( f_j u \times (u \times \partial ^{h}u) \right) , \partial ^{h} \left( f_j u \times \partial ^{h}u \right) \right\rangle \\&\quad = \left\langle (\partial ^{h} f_j) u^+ \times (u^+ \times \partial ^{h}u^+) + f_j \partial ^{h}(u \times (u \times \partial ^{h}u)), (\partial ^{h} f_j) u^+ \times \partial ^{h}u^+ +f_j (u \times \Delta ^h u)^+ \right\rangle \\&\quad = (\partial ^{h} f_j) f_j \left\langle u^+ \times (u^+ \times \partial ^{h}u^+), (u \times \Delta ^h u)^+ \right\rangle \\&\qquad + \left\langle f_j \partial ^{h}u \times (u^+ \times \partial ^{h}u^+) + f_j u \times (u \times \Delta ^h u)^+, (\partial ^{h} f_j) u^+ \times \partial ^{h}u^+ +f_j (u \times \Delta ^h u)^+ \right\rangle \\&\quad = (\partial ^{h} f_j) f_j \left\langle (u^+-u) \times (u^+ \times \partial ^{h}u^+), (u \times \Delta ^h u)^+ \right\rangle \\&\qquad + f_j^2 \left\langle \partial ^{h}u \times (u^+ \times \partial ^{h}u^+), u^+ \times \Delta ^h u^+ \right\rangle . \end{aligned}$$

Since $\sum _j q_j^2 |f_j| |\partial ^h f_j|(x) \le C_\kappa ^2$ for all $x \in {\mathbb {Z}}_h$, we have

$$\begin{aligned} T_{33}(s)= & \gamma h \sum _x \sum _j q_j^2 \left\langle \partial ^{h} \left( f_j m^h \times m^h \times \partial ^{h}m^h \right) , \partial ^{h}\left( f_j m^h \times \partial ^{h}m^h \right) \right\rangle (s,x) \nonumber \\\le & |\gamma | h \sum _x \sum _j q_j^2 |\partial ^{h} f_j^-| \ |f_j^-| \left( \frac{1}{\varepsilon ^2} |\partial ^{h}m^h|^2 + \varepsilon ^2 |m^h \times \Delta ^h m^h|^2 \right) (s,x) \nonumber \\ & + \gamma h \sum _x \sum _j q_j^2 (f_j^-)^2 \left\langle m^h, (\partial ^h m^h)^- \right\rangle \left\langle m^h \times \partial ^{h}m^h, \Delta ^h m^h \right\rangle (s,x) \nonumber \\\le & |\gamma | C_\kappa ^2 \left( \frac{1}{\varepsilon ^2} |\partial ^{h}m^h|^2_{{\mathbb {L}}_h^2}(s) + \varepsilon ^2 |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2}(s) \right) - T_{22d}(s), \end{aligned}$$

(43)

where $T_{22d}(s)$ is given in (29).

$\underline{\text { An estimate on } \,\,T_1 + T_2 + T_3:}$

$$\begin{aligned} T_1 + T_2 + T_3&= T_1 + T_{21} + T_{22a} + T_{22b} + T_{22c} + T_{22d} + T_{31} + T_{32} + T_{33}, \end{aligned}$$

where by (30) and (41),

$$\begin{aligned} T_{22a}(s) + T_{22c}(s) + T_{31}(s)&\le \left( \frac{1}{2} \gamma ^2 +1 \right) C_\kappa ^2 |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2}(s) + C_\kappa ^2 |\partial ^{h} m^h|^2_{{\mathbb {L}}_h^2}(s), \end{aligned}$$

and by (43),

$$\begin{aligned} T_{22d}(s) + T_{33}(s)&\le |\gamma | C_\kappa ^2 \left( \frac{1}{\varepsilon ^2} |\partial ^{h}m^h|^2_{{\mathbb {L}}_h^2}(s) + \varepsilon ^2 |m^h \times \Delta ^h m^h|^2_{{\mathbb {L}}_h^2}(s) \right) . \end{aligned}$$

Then, by (27), (28), (33) and (42),

$$\begin{aligned} T_1(s) + T_2(s) + T_3(s) \le \frac{1}{2}C_{1,\varepsilon } \ |\partial ^h m^h(s)|^2_{{\mathbb {L}}_h^2} +\left( \frac{1}{2} C_{2,\varepsilon } - \alpha \right) \ |m^h(s) \times \Delta ^h m^h(s)|^2_{{\mathbb {L}}_h^2}, \nonumber \\ \end{aligned}$$

(44)

where

$$\begin{aligned} \begin{aligned} C_{1,\varepsilon }&:= \frac{1}{2\varepsilon ^2}\left[ \left( \gamma ^2 + 1 \right) ^2 C_\kappa ^4 + 4|\gamma | C_\kappa ^2 + 2C_v^2 (1+\gamma ^2) \right] + 2\left( 1 + \gamma ^2 \right) C_\kappa ^2, \\ C_{2,\varepsilon }&:= \left( 4 \gamma ^2 +2 \right) C_\kappa ^2 + \varepsilon ^2 \left( 3 + 2|\gamma | C_\kappa ^2 \right) . \end{aligned} \end{aligned}$$

(45)

$\underline{\text { Uniform estimate of } \,\, \partial ^h m^h }:$

Using (24) and (44), we have

$$\begin{aligned} \begin{aligned}&|\partial ^h m^h(t)|^{2}_{{\mathbb {L}}_h^2} + (2\alpha -C_{2,\varepsilon }) \int _0^t | m^h(s) \times \Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \\&\quad \le |\partial ^h m_0^h|^2_{{\mathbb {L}}_h^2} + C_{1,\varepsilon } \int _0^t |\partial ^h m^h(s)|^2_{{\mathbb {L}}_h^2} \ \textrm{d}s + 2\sup _{t \in [0,T]} |M^h(t)|. \end{aligned} \end{aligned}$$

(46)

Taking a sufficiently small $\varepsilon$ such that $\frac{1}{2}\varepsilon ^2 \left( 3 + 2|\gamma | C_\kappa ^2 \right) < \delta$, we have from (21):

$$\begin{aligned} 2\alpha -C_{2,\varepsilon } > 0. \end{aligned}$$

(47)

Then, for $p \ge 1$ and $q = \frac{p}{p-1}$,

$$\begin{aligned}&{\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^h m^h(t)|^{2p}_{{\mathbb {L}}_h^2} + (2\alpha - C_{2,\varepsilon })^p \left( \int _0^T |m^h(s) \times \Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \right) ^p \right] \\&\quad \le {\mathbb {E}}\left[ \left( \sup _{t \in [0,T]} |\partial ^h m^h(t)|^{2}_{{\mathbb {L}}_h^2} + (2\alpha - C_{2,\varepsilon }) \int _0^T |m^h(s) \times \Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \right) ^p \right] \\&\quad \le {\mathbb {E}}\left[ \left( |\partial ^h m_0^h|^2_{{\mathbb {L}}_h^2} + C_{1,\varepsilon } \int _0^T |\partial ^h m^h(s)|^2_{{\mathbb {L}}_h^2} \ \textrm{d}s + 2 \sup _{t \in [0,T]} |M^h(t)| \right) ^p \right] \\&\quad \le (2^{p-1})^2 {\mathbb {E}}\left[ |\partial ^h m_0^h|_{{\mathbb {L}}_h^2}^{2p} + C_{1,\varepsilon }^p T^{\frac{p}{q}} \int _0^T \sup _{t \in [0,s]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \ \textrm{d}s \right] \\&\qquad + 2^{p-1} {\mathbb {E}}\left[ 2^p \sup _{t \in [0,T]} |M^h(t)|^p \right] \\&\le 4^{p-1} {\mathbb {E}}\left[ |\partial ^h m_0^h|_{{\mathbb {L}}_h^2}^{2p} + C_{1,\varepsilon }^p T^{\frac{p}{q}} \int _0^T \sup _{t \in [0,s]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \ \textrm{d}s \right] \\&\qquad + 4^{p-1} b_p (1+|\gamma |)^p C_\kappa ^p \ {\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} + \left( \int _0^T |m^h(t) \times \Delta ^h m^h(t)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}t \right) ^p \right] , \end{aligned}$$

where the second inequality holds by (46) and the last inequality holds by Lemma 3.4. Then, by the definitions of $N_{1,p}$ and $N_{2,p}$,

$$\begin{aligned} \begin{aligned}&N_{1,p} {\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \right] + N_{2,p} {\mathbb {E}}\left[ \left( \int _0^T |m^h(s) \times \Delta ^h m^h(s) |_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \right) ^p \right] \\&\quad \le 4^{p-1} {\mathbb {E}}\left[ |\partial ^h m_0^h|_{{\mathbb {L}}_h^2}^{2p} + C_{1,\varepsilon }^p T^\frac{p}{q} \int _0^T \sup _{t \in [0,s]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \ \textrm{d}s \right] . \end{aligned} \end{aligned}$$

(48)

Hence, by (21) and (48),

$$\begin{aligned} & {\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \right] \le N_{1,p}^{-1} 4^{p-1} \nonumber \\ & \quad \left( {\mathbb {E}}\left[ |\partial ^h m_0^h|_{{\mathbb {L}}_h^2}^{2p} \right] + C_{1,\varepsilon }^p T^\frac{p}{q} {\mathbb {E}}\left[ \int _0^T \sup _{t \in [0,s]} |\partial ^h m^h(t)|_{{\mathbb {L}}_h^2}^{2p} \ \textrm{d}s \right] \right) . \end{aligned}$$

(49)

By Fubini’s theorem and Grönwall’s inequality,

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t \in [0,T]} |\partial ^{+h} m^h(t) |_{{\mathbb {L}}_h^2}^{2p} \right] \le N_{1,p}^{-1} 4^{p-1} K_0^{2p} \exp \left( \int _0^T N_{1,p}^{-1} 4^{p-1} C_{1,\varepsilon }^p T^\frac{p}{q} \ \textrm{d}t \right) = K_{1,p},\nonumber \\ \end{aligned}$$

(50)

where $K_{1,p}$ depends on $p, C_v, C_\kappa , \varepsilon , T$ and $K_0$, but not on h, proving (22).

Finally, by (21), (48) and (50),

$$\begin{aligned} {\mathbb {E}}\left[ \left( \int _0^T |m^h(s) \times \Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \right) ^p \right]&\le N_{2,p}^{-1} 4^{p-1} \left( K_0^{2p} + C_{1,\varepsilon }^p T^{\frac{p}{q}+1} K_{1,p} \right) = K_{2,p}, \end{aligned}$$

where $K_{2,p}$ depends on $p, C_v, C_\kappa , \varepsilon , T$ and $K_0$, but not on h, proving (23). $\square$

Remark 3.6

Fix $p \in [1,\infty )$, if

$$\begin{aligned} C_\kappa \le \frac{\alpha -\delta }{1+2\gamma ^2 + 2^{1-\frac{2}{p}} b_p^{\frac{1}{p}} (1+|\gamma |)} \wedge \frac{1}{4 b_p^{\frac{1}{p}} (1+|\gamma |)} \wedge 1-\delta , \end{aligned}$$

then the assumption (21) of Lemma 3.5 is satisfied.

Lemma 3.7

For any $p \in [1, \infty )$, under the conditions of Lemma 3.5, there exists a constant $K_{3,p}$ independent of h such that

$$\begin{aligned} {\mathbb {E}}\left[ \left( \int _0^T |\Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2 \ \textrm{d}s \right) ^p \right] \le K_{3,p}. \end{aligned}$$

(51)

Proof

Since $|m^h(s,x)|=1$ for all $(s,x) \in [0,T] \times {\mathbb {Z}}_h$, we have

$$\begin{aligned} |\Delta ^h m^h(s,x)|^2 = |m^h(s,x) \times \Delta ^h m^h(s,x)|^2 + \langle m^h(s,x), \Delta ^h m^h(s,x) \rangle ^2. \end{aligned}$$

Then,

$$\begin{aligned} |\Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2&= h \sum _{x \in {\mathbb {Z}}_h} \left( |m^h(s,x) \times \Delta ^h m^h(s,x)|^2 + \langle m^h(s,x), \Delta ^h m^h(s,x) \rangle ^2 \right) \\&= |m^h(s) \times \Delta ^h m^h(s)|^2_{{\mathbb {L}}_h^2} + h \sum _{x \in {\mathbb {Z}}_h} \left( \frac{1}{2} (|\partial ^h m^h(s,x)|^2 + |(\partial ^h m^h)^-(s,x)|^2) \right) ^2 \\&\le |m^h(s) \times \Delta ^h m^h(s)|^2_{{\mathbb {L}}_h^2} + |\partial ^{h} m^h(s)|^4_{{\mathbb {L}}_h^4}, \end{aligned}$$

where

$$\begin{aligned} |\partial ^{h} m^h(s)|^4_{{\mathbb {L}}_h^4}&\le |\partial ^{h} m^h(s)|^2_{{\mathbb {L}}_h^\infty } \ |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^2}^2. \end{aligned}$$

Applying Lemma A.1 on $\partial ^{h} m^h(s)$,

$$\begin{aligned} |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^\infty } \le C |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^2}^{\frac{1}{2}} |\partial ^{h}(\partial ^{h} m^h(s)) |_{{\mathbb {L}}_h^2}^{\frac{1}{2}}, \end{aligned}$$

(52)

where $|\partial ^{h}(\partial ^{h} m^h(s)) |_{{\mathbb {L}}_h^2} = |\Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}$. Thus,

$$\begin{aligned} \begin{aligned} |\partial ^{h} m^h(s)|^4_{{\mathbb {L}}_h^4}&\le C^2 |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^2}^3 \ |\Delta ^h m^h(s)|_{{\mathbb {L}}_h^2} \\&\le \frac{1}{2} C^4 |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^2}^6 + \frac{1}{2}|\Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2. \end{aligned} \end{aligned}$$

(53)

We have

$$\begin{aligned} |\Delta ^h m^h(s)|_{{\mathbb {L}}_h^2}^2&\le 2|m^h(s) \times \Delta ^h m^h(s)|^2_{{\mathbb {L}}_h^2} + C^4 |\partial ^{h} m^h(s)|_{{\mathbb {L}}_h^2}^6. \end{aligned}$$

Then, by Lemma 3.5,

$$\begin{aligned} {\mathbb {E}}\left[ \left( \int _0^T |\Delta ^h m^h(t) |^2_{{\mathbb {L}}_h^2} \ \textrm{d}t\right) ^p \right]&\le 2^{p(p-1)}K_{2,p} + 2^{p-1} C^{4p} K_{1, 3p}T^p = K_{3,p}, \end{aligned}$$

proving (51). $\square$

4 Quadratic interpolation for the solution $m^h$ of (11)

4.1 Interpolations

For any fixed $h>0$, let $x_k = kh \in {\mathbb {Z}}_h$, for $k \in {\mathbb {Z}}$. We introduce interpolations of discrete functions defined on ${\mathbb {Z}}_h$ to functions defined on ${\mathbb {R}}$.

Given $u: {\mathbb {Z}}_h \rightarrow {\mathbb {R}}^3$, let $\overline{u}: {\mathbb {R}}\rightarrow {\mathbb {R}}^3$ denote a quadratic interpolation of u, given by

$$\begin{aligned} \overline{u}(x) = \frac{1}{2} \left( u(x_k) + u(x_{k-1}) \right) + \partial ^{h} u^-(x_k) (x-x_k) + \frac{1}{2} \Delta ^h u(x_k) (x-x_k)^2, \end{aligned}$$

(54)

for $x \in [x_k,x_{k+1})$, $k \in {\mathbb {Z}}$, where $\overline{u}$ is continuously differentiable with

$$\begin{aligned} \begin{aligned} D \overline{u}(x)&= \partial ^{h} u^-(x_k) + \Delta ^h u(x_k) (x-x_k), & \quad x \in [x_k, x_{k+1}), \\ D^2 \overline{u}(x)&= \Delta ^h u(x_k), & \quad x \in (x_k,x_{k+1}). \end{aligned} \end{aligned}$$

(55)

Let $\widehat{u}: {\mathbb {R}}\rightarrow {\mathbb {R}}^3$ denote the piecewise constant interpolation of u, given by

$$\begin{aligned} \widehat{u}(x) = u(x_k), \quad x \in [x_k, x_{k+1}), \end{aligned}$$

(56)

for any $k \in {\mathbb {Z}}$. In terms of $\widehat{u}$, we can express $\overline{u}$ as

$$\begin{aligned} \begin{aligned} \overline{u}(x)&= \frac{1}{2} \left( \widehat{u}(x) + \widehat{u}^-(x) \right) + \partial ^{h} \widehat{u}^-(x) (x-x_k) + \frac{1}{2} \Delta ^h \widehat{u}(x) (x-x_k)^2 \\&= \widehat{u}(x) + \left( D\overline{u}(x) - D^2\overline{u}(x) (x-x_k) \right) \left( x-x_k - \frac{h}{2} \right) + \frac{1}{2} D^2\overline{u}(x) (x-x_k)^2, \end{aligned} \end{aligned}$$

(57)

for $x \in [x_k,x_{k+1})$, $k \in {\mathbb {Z}}$.

We collect estimates of $\overline{u}$ and $\widehat{u}$ in terms of u in the following remark.

Remark 4.1

Let $u: {\mathbb {Z}}_h \rightarrow {\mathbb {R}}^3$. Then

$$\begin{aligned}&|\widehat{u}|_{{\mathbb {L}}^\infty } = |u|_{{\mathbb {L}}_h^\infty }, \\&|\widehat{u}|_{{\mathbb {L}}^2_w} \le |\widehat{u}|_{{\mathbb {L}}^2} = |u|_{{\mathbb {L}}_h^2}, \quad w > 0, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}&|\overline{u}|_{{\mathbb {L}}^\infty } \le 5|u|_{{\mathbb {L}}_h^\infty }, \\&|D\overline{u}|_{{\mathbb {L}}^2} \le 3|\partial ^{h} u|_{{\mathbb {L}}_h^2}, \\&|D^2\overline{u}|_{{\mathbb {L}}^2} = |\Delta ^h u|_{{\mathbb {L}}_h^2}, \\&|D\overline{u}|_{{\mathbb {L}}^4}^4 \le \frac{1}{2} C^4 |D\overline{u}|_{{\mathbb {L}}^2}^6 + \frac{1}{2} |D^2 \overline{u}|_{{\mathbb {L}}^2}^2. \end{aligned} \end{aligned}$$

(58)

4.2 An equation and estimates for $\overline{m}^h$

4.2.1 Equation for $\overline{m}^h$

Since $m^h$ is the solution of the semi-discrete scheme (11), the piecewise constant interpolation $\widehat{m}^h$ satisfies

$$\begin{aligned} \widehat{m}^h(t)&= \widehat{m}_0^h + \int _0^t \left( F^h_{\widehat{v}}(\widehat{m}^h(s)) + \frac{1}{2} S^h_{\widehat{\kappa }}(\widehat{m}^h(s)) \right) \textrm{d}s + \int _0^t G^h(\widehat{m}^h(s)) \ \textrm{d}\widehat{W}^h(s), \nonumber \\ \end{aligned}$$

(59)

where $F^h_{\widehat{v}}$ and $S^h_{\widehat{\kappa }}$ are defined as in (10) but with $\widehat{v}$, $\widehat{\kappa }$ and $\widehat{\kappa \kappa '}$ in place of v, $\kappa$ and $\kappa \kappa '$, respectively, and

$$\begin{aligned} \widehat{W}^h(t) := \sum _{j=1}^\infty q_j W_j(t) \widehat{f}_j, \quad t \in [0,T]. \end{aligned}$$

In particular, for every fixed $h>0$, $m^h \in {\mathcal {C}}([0,T]; {\mathbb {E}}_h)$ and $\widehat{m}^h \in {\mathcal {C}}([0,T]; {\mathbb {L}}^2_w \cap \mathring{{\mathbb {H}}}^1)$ for $w \ge 1$.

In order to obtain an equation for $\overline{m}^h$, by using (57) we note that

$$\begin{aligned}&\widehat{m}^h = \overline{m}^h - R_0\overline{m}^h, \quad \partial ^{h} \widehat{m}^h = D\overline{m}^h - R_1\overline{m}^h, \\&\widehat{m}^h \times \partial ^{h} \widehat{m}^h = \overline{m}^h \times D\overline{m}^h - P_1\overline{m}^h, \quad \widehat{m}^h \times \\&\left( \widehat{m}^h \times \partial ^{h} \widehat{m}^h \right) = \overline{m}^h \times \left( \overline{m}^h \times D\overline{m}^h \right) - P_2\overline{m}^h, \\&|\widehat{m}^h \times \partial ^{h} \widehat{m}^h|^2 \widehat{m}^h = \left| \overline{m}^h \times D\overline{m}^h \right| ^2 \overline{m}^h - P_3\overline{m}^h, \\&\partial ^h \widehat{m}^h \times \left( \widehat{m}^h \times \partial ^h \widehat{m}^h \right) = D\overline{m}^h \times \left( \overline{m}^h \times D\overline{m}^h \right) - P_4\overline{m}^h, \\&\langle \widehat{m}^h, \partial ^h \widehat{m}^{h-} \rangle \ \widehat{m}^h \times \partial ^h \widehat{m}^h = \langle \overline{m}^h, D\overline{m}^h \rangle \ \overline{m}^h \times D\overline{m}^h - P_5\overline{m}^h, \\&\widehat{m}^h \times \Delta ^h \widehat{m}^h = \overline{m}^h \times D^2\overline{m}^h - Q_1\overline{m}^h, \quad \widehat{m}^h \\&\times \left( \widehat{m}^h \times \Delta ^h \widehat{m}^h \right) = \overline{m}^h \times \left( \overline{m}^h \times D^2\overline{m}^h \right) - Q_2\overline{m}^h, \end{aligned}$$

where for $u: {\mathbb {R}}\rightarrow {\mathbb {R}}^3$ with well-defined weak derivatives,

$$\begin{aligned} \begin{aligned} R_0u(x)&:= \left( Du(x) - D^2u(x) (x-x_k) \right) \left( x-x_k - \frac{h}{2} \right) + \frac{1}{2} D^2u(x) (x-x_k)^2, \\ R_1u(x)&:= D^2u(x) (x-x_k-h), \\ P_1u(x)&:= R_0u(x) \times Du(x) + (u-R_0u)(x) \times R_1 u(x), \end{aligned} \end{aligned}$$

(60)

and

$$\begin{aligned} P_2u(x):= & \left[ u \times P_1u + R_0u \times \left( u \times Du - P_1u \right) \right] (x), \nonumber \\ P_3u(x):= & \left[ \left\langle 2 u \times Du - P_1u, P_1u \right\rangle u + \left| u \times Du - P_1u \right| ^2 R_0u \right] (x), \nonumber \\ P_4u(x):= & \left[ R_1 u \times (u \times Du) + (Du - R_1u) \times P_1u \right] (x), \nonumber \\ P_5u(x):= & \left( \langle R_0 u(x), Du(x) \rangle + \langle (u -R_0u)(x), D^2u(x) (x-x_k) \rangle \right) u(x) \times Du(x) \nonumber \\ & + \langle (u-R_0u)(x), Du(x) - D^2u(x)(x-x_k) \rangle P_1 u(x), \nonumber \\ Q_1u(x):= & R_0u(x) \times D^2u(x), \nonumber \\ Q_2u(x):= & \left[ u \times Q_1u + R_0u \times \left( (u - R_0u) \times D^2u \right) \right] (x), \end{aligned}$$

(61)

for $x \in [x_k,x_{k+1})$, $k \in {\mathbb {Z}}$.

Moreover, for $u: [0,T] \times {\mathbb {R}}\rightarrow {\mathbb {R}}^3$, define

$$\begin{aligned} \begin{aligned} F_{\widehat{v}}(u)&:= - u \times \left( D^2u + \alpha u \times D^2u \right) + \widehat{v} \left( u \times \left( u \times Du \right) + \gamma u \times Du \right) , \\ S_{\widehat{\kappa }}(u)&:= \frac{1}{2}\left( (\widehat{\kappa }^-)^2 + \widehat{\kappa }^2 \right) \left( (\gamma ^2 - 1) u \times \left( u \times D^2 u\right) - 2 \gamma u \times D^2 u \right) \\&\quad - \widehat{\kappa }^2 \left( \gamma ^2 Du \times (u \times Du) + |u \times Du|^2 u \right) + 2 \gamma (\widehat{\kappa }^2)^- \langle u, Du \rangle u \times Du \\&\quad + \widehat{\kappa \kappa '} \left[ (\gamma ^2 -1) u \times (u \times Du) - 2 \gamma u \times Du \right] . \end{aligned} \end{aligned}$$

(62)

By (59), we arrive at the equation of $\overline{m}^h$. For $t \in [0,T]$,

$$\begin{aligned} \begin{aligned} \overline{m}^h(t)&= \widehat{m}_0^h + \int _0^t F_{\widehat{v}}(\overline{m}^h(s)) \ \textrm{d}s + \frac{1}{2} \int _0^t S_{\widehat{\kappa }}(\overline{m}^h(s)) \ \textrm{d}s + \int _0^t G(\overline{m}^h(s)) \ \textrm{d}\widehat{W}^h(s) \\&\quad + R_0 \overline{m}^h(t) + \int _0^t \left( -\widehat{v}(\gamma P_1 + P_2) \overline{m}^h + Q_1\overline{m}^h +\alpha Q_2\overline{m}^h \right) (s) \ \textrm{d}s \\&\quad + \frac{1}{4} \int _0^t \left( (\widehat{\kappa }^2)^- + \widehat{\kappa }^2 \right) \left[ 2 \gamma Q_1\overline{m}^h - (\gamma ^2-1) Q_2\overline{m}^h \right] (s) \ \textrm{d}s \\&\quad + \frac{1}{2} \int _0^t \left( \widehat{\kappa }^2 (P_3\overline{m}^h + \gamma ^2 P_4\overline{m}^h) - 2 \gamma (\widehat{\kappa }^2)^- P_5\overline{m}^h \right) (s) \ \textrm{d}s \\&\quad + \frac{1}{2}\int _0^t \widehat{\kappa \kappa '} \left[ 2\gamma P_1 \overline{m}^h - (\gamma ^2-1) P_2\overline{m}^h \right] (s) \ \textrm{d}s \\&\quad - \int _0^t (\gamma P_1 + P_2) \overline{m}^h(s) \ \textrm{d}\widehat{W}^h(s). \end{aligned} \end{aligned}$$

(63)

4.2.2 Estimates for $\overline{m}^h$

For $p \in [1, \infty )$ and $w \ge 1$, we deduce from Lemmata 3.5, 3.7 and Remark 4.1:

$$\begin{aligned} \sup _{t \in [0,T]} |\overline{m}^h(t)|_{{\mathbb {L}}^\infty }^p \le 5^p, \quad {\mathbb {P}}\text {-a.s.} \end{aligned}$$

(64)

and

$$\begin{aligned} & {\mathbb {E}}\left[ \sup _{t \in [0,T]} |D\overline{m}^h(t)|_{{\mathbb {L}}^2}^{2p} + \left( \int _0^T \left( |\overline{m}^h(t)|_{{\mathbb {L}}^p_w}^p + |D\overline{m}^h(t)|_{{\mathbb {L}}^4}^4 + |D^2 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \right) \textrm{d}t \right) ^p \right] \nonumber \\ & \quad \le C(p,T,w), \end{aligned}$$

(65)

for some constant C(p, T, w) .

Results of convergence of $R_0 \overline{m}^h$, $R_1 \overline{m}^h$, $P_1 \overline{m}^h,\ldots , P_5 \overline{m}^h$ and $Q_1 \overline{m}^h$, $Q_2 \overline{m}^h$ are proved in the following lemma.

Lemma 4.2

For $f = R_1$, $P_1$ or $P_2$,

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}\left[ \sup _{t \in [0,T]} |R_0 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 + \int _0^T |f \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] = 0. \end{aligned}$$

Moreover, for $f=P_3$, $P_4$, $P_5$, $Q_1$ or $Q_2$, for any measurable process $\varphi \in L^4(\Omega ; L^4(0,T; {\mathbb {L}}^4))$,

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}\left[ \int _0^T \left\langle f\overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \right] = 0. \end{aligned}$$

Proof

By construction, $|x-x_k| < h$ for $x \in [x_k, x_{k+1})$, $k \in {\mathbb {Z}}$, and $\sup _{t \in [0,T]} |\overline{m}^h(t)|_{{\mathbb {L}}^\infty } \le 5$, ${\mathbb {P}}$-a.s. Thus, for $p \in [1, \infty )$, there is a constant $C_{R_0}$ independent of h such that

$$\begin{aligned} \sup _{t \in [0,T]} |R_0\overline{m}^h(t)|_{{\mathbb {L}}^\infty }^p \le C_{R_0}^p, \quad {\mathbb {P}}\text {-a.s.} \end{aligned}$$

Using (55), we can often rewrite $R_0, \ldots , Q_2$ in terms of $\widehat{m}^h$ to simplify the estimates.

$\underline{\text { An estimate on } \,\, R_0\overline{m}^h:}$

$$\begin{aligned} R_0 \overline{m}^h(t,x) = \partial ^h \widehat{m}^h(t,x-h) \left( x-x_k- \frac{h}{2}\right) + \frac{1}{2} \Delta ^h \widehat{m}(t,x) (x-x_k)^2, \quad x \in [x_k, x_{k+1}], \end{aligned}$$

which implies

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t \in [0,T]} |R_0 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \right]&\le \frac{h^2}{2}{\mathbb {E}}\left[ \sup _{t \in [0,T]} \left( |\partial ^h \widehat{m}^h(t)|_{{\mathbb {L}}^2}^2 + |\partial ^h \widehat{m}^h(t) - \partial ^h \widehat{m}^{h-}(t)|_{{\mathbb {L}}^2}^2 \right) \right] . \nonumber \\ \end{aligned}$$

(66)

The expectation on the right-hand side of (66) is bounded by Lemma 3.5; thus, the left-hand side converges to 0 as $h \rightarrow 0$. As a result,

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T |R_0\overline{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \le {\mathbb {E}}\left[ C_{R_0}^2 T \sup _{t \in [0,T]} |R_0 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \right] {\mathop {\rightarrow }\limits ^{h \rightarrow 0}} 0. \end{aligned}$$

(67)

$\underline{\text { An estimate on } \,\, R_1\overline{m}^h:}$

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T |R_1 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right]&\le \frac{h^2}{2}{\mathbb {E}}\left[ \int _0^T | D^2 \overline{m}^h(t) |_{{\mathbb {L}}^2}^2 \ \textrm{d}x \ \textrm{d}t \right] , \end{aligned}$$

implying $R_1 \overline{m}^h \rightarrow 0$ in $L^2(\Omega ; L^2(0,T; {\mathbb {L}}^2))$ by (65).

$\underline{\text { An estimate on } \,\, P_1\overline{m}^h:}$

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T |P_1 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right]&= {\mathbb {E}}\left[ \int _0^T |R_0 \overline{m}^h(t) \times D \overline{m}^h(t) + \widehat{m}^h(t) \times R_1\overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \\&\le 2 \left( {\mathbb {E}}\left[ \int _0^T |R_0 \overline{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}\left[ \int _0^T |D \overline{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2} \\&\quad + 2 {\mathbb {E}}\left[ \int _0^T |R_1\overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] , \end{aligned}$$

where $D\overline{m}^h \in L^4(\Omega ; L^4(0,T; {\mathbb {L}}^4))$ for all $h > 0$ by (65). Then, by the $L^4$-convergence of $R_0\overline{m}^h$ in (67) and the $L^2$-convergence of $R_1\overline{m}^h$, we have $P_1 \overline{m}^h \rightarrow 0$ in $L^2(\Omega ; L^2(0,T; {\mathbb {L}}^2))$.

$\underline{\text { An estimate on } \,\, P_2\overline{m}^h:}$

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T |P_2 \overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \\&\quad = {\mathbb {E}}\left[ \int _0^T |\widehat{m}^h(t)\times P_1 \overline{m}^h(t) + R_0 \overline{m}^h(t) \times (\overline{m}^h(t) \times D \overline{m}^h(t))|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \\&\quad \le 2 {\mathbb {E}}\left[ \int _0^T |\widehat{m}^h(t) \times P_1\overline{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \\&\qquad + 2 \left( {\mathbb {E}}\left[ \int _0^T |R_0 \overline{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}\left[ \int _0^T |\overline{m}^h(t) \times D \overline{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

which implies that $P_2 \overline{m}^h \rightarrow 0$ in $L^2(\Omega ; L^2(0,T; {\mathbb {L}}^2))$ by (64) and (65) together with the convergences of $P_1\overline{m}^h$ and $R_0\overline{m}^h$.

$\underline{\text { An estimate on } \,\, P_3 \overline{m}^h:}$

$$\begin{aligned} P_3 \overline{m}^h&= |\widehat{m}^h \times \partial ^h \widehat{m}^h|^2 R_0\overline{m}^h + \left\langle \overline{m}^h \times D\overline{m}^h + \widehat{m}^h \times \partial ^h \widehat{m}^h, P_1\overline{m}^h \right\rangle \overline{m}^h \\&=: P_{31} \overline{m}^h + P_{32} \overline{m}^h. \end{aligned}$$

Then for $\varphi \in L^4(\Omega ; L^4(0,T; {\mathbb {L}}^4))$,

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T \left\langle P_{31} \overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right]&\le \left( {\mathbb {E}}\left[ \int _0^T |\widehat{m}^h(t) \times \partial ^h \widehat{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{2} \\&\quad \times \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4} \left( {\mathbb {E}}\left[ \int _0^T |R_0\overline{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4}, \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \left\langle P_{32}\overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \textrm{d}t \right] \\&\quad \le 5\left( {\mathbb {E}}\left[ \int _0^T\left| \overline{m}^h(t) \times D\overline{m}^h(t) + \widehat{m}^h(t) \times \partial ^{h} \widehat{m}^h(t) \right| ^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4} \left( {\mathbb {E}}\left[ \int _0^T |P_1\overline{m}^h(t)|^2_{{\mathbb {L}}^2} \ \textrm{d}t \right] \right) ^\frac{1}{2}. \end{aligned}$$

By Lemmata 3.5 and 3.7, (64), (65) and the property of $\varphi$, the expectations on the right-hand side of the two inequalities above are finite. Then by the convergences of $R_0\overline{m}^h$ and $P_1\overline{m}^h$, we obtain the weak convergence of $P_3\overline{m}^h$ as desired.

$\underline{\text { An estimate on } \,\, P_4\overline{m}^h:}$

$$\begin{aligned} P_4 \overline{m}^h&= R_1\overline{m}^h \times (\overline{m}^h \times D\overline{m}^h) + \partial ^h\widehat{m}^h \times P_1\overline{m}^h \\&=: P_{41}\overline{m}^h + P_{42}\overline{m}^h. \end{aligned}$$

We have

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T \left\langle P_{41} \overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right]&\le 5 \left( {\mathbb {E}}\left[ \int _0^T |R_1\overline{m}^h(t)|^2_{{\mathbb {L}}^2} \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}\left[ \int _0^T |D\overline{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\quad \times \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4}, \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \left\langle P_{42} \overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}\left[ \int _0^T |P_1\overline{m}^h(t)|^2_{{\mathbb {L}}^2} \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}\left[ \int _0^T |\partial ^h\widehat{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4}. \end{aligned}$$

Using (65), (53) and the convergences of $R_1 \overline{m}^h$ and $P_1 \overline{m}^h$, the right-hand side of each of the two inequalities above converges to 0 as $h \rightarrow 0$.

$\underline{\text { An estimate on } \,\, P_5\overline{m}^h:}$

$$\begin{aligned} P_5\overline{m}^h(x)&= \langle R_0 \overline{m}^h(x), D\overline{m}^h(x) \rangle \overline{m}^h(x) \times D\overline{m}^h(x) + (x-x_k) \langle \widehat{m}^h(x), D^2\overline{m}^h(x) \rangle \overline{m}^h(x) \\&\quad \times D\overline{m}^h(x) + \langle \widehat{m}^h(x), \partial ^h \widehat{m}^{h-} (x) \rangle P_1 u(x) \\&=: P_{51}\overline{m}^h(x) + P_{52}\overline{m}^h(x) + P_{53}\overline{m}^h(x), \quad x \in [x_k, x_{k+1}). \end{aligned}$$

By Lemma 2.4 and (64), we have

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \left\langle P_{51}\overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le 5 \left( {\mathbb {E}}\left[ \int _0^T |R_0\overline{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}\left[ \int _0^T |D\overline{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{2} \\&\qquad \times \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4}, \\&{\mathbb {E}}\left[ \int _0^T \left\langle P_{52}\overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \le 5h \left( {\mathbb {E}}\left[ \int _0^T |D^2\overline{m}^h(t)|^2_{{\mathbb {L}}^2} \ \textrm{d}t \right] \right) ^\frac{1}{2} \\&\qquad \times \left( {\mathbb {E}}\left[ \int _0^T |D\overline{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4}, \end{aligned}$$

and

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \left\langle P_{53}\overline{m}^h(t), \varphi (t) \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}\left[ \int _0^T |P_1\overline{m}^h(t)|^2_{{\mathbb {L}}^2} \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}\left[ \int _0^T |\partial ^h \widehat{m}^h(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t\right] \right) ^\frac{1}{4}. \end{aligned}$$

Similarly, by Lemma 3.5, (65) and the convergences of $R_0\overline{m}^h$ and $P_1\overline{m}^h$, the right-hand side of each of the inequalities above converges to 0 as $h \rightarrow 0$.

$\underline{\text { An estimate on } \,\, Q_1\overline{m}^h:}$

$$\begin{aligned} Q_1\overline{m}^h(t,x)&= R_0 \overline{m}^h(t,x) \times D^2 \overline{m}^h(t,x) \\&= \partial ^h \widehat{m}^{h-}(t,x) \times \Delta ^h \widehat{m}^h(t,x) \left( x-x_k - \frac{h}{2} \right) , \quad x \in [x_k,x_{k+1}). \end{aligned}$$

Thus,

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \langle Q_1\overline{m}^h(t), \varphi (t) \rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \frac{h}{2} {\mathbb {E}}\left[ \int _0^T \int _{\mathbb {R}}|\partial ^h \widehat{m}^{h-}(t,x)| |\Delta ^h \widehat{m}^h(t,x)| |\varphi (t,x)| \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le \frac{h}{2} \left( {\mathbb {E}}\left[ \int _0^T |\partial ^{h} \widehat{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}\left[ \int _0^T |\Delta ^h \widehat{m}^h(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4}, \end{aligned}$$

where the three expectation terms on the right-hand side are finite, proving that the right-hand side converges to 0 as $h \rightarrow 0$.

$\underline{\text { An estimate on } \,\, Q_2\overline{m}^h:}$

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \langle Q_2\overline{m}^h(t), \varphi (t) \rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad = {\mathbb {E}}\left[ \int _0^T \langle Q_1\overline{m}^h(t), \varphi (t) \times \overline{m}^h(t) \rangle _{{\mathbb {L}}^2} \ \textrm{d}t + \int _0^T \langle R_0\overline{m}^h(t), (\widehat{m}^h(t) \times \Delta ^h \widehat{m}^h(t)) \times \varphi (t) \rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}\left[ \int _0^T \langle Q_1\overline{m}^h(t), \varphi (t) \times \overline{m}^h(t) \rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\qquad + \left( {\mathbb {E}}\left[ \int _0^T |R_0 \overline{m}^h(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^{\frac{1}{4}} \left( \int _0^T |\widehat{m}^h(t) \times \Delta ^h \widehat{m}^h(t) |_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right) ^\frac{1}{2}\\&\qquad \left( \int _0^T |\varphi (t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right) ^\frac{1}{4}, \end{aligned}$$

where on the right-hand side, the first term converges to 0 as $h \rightarrow 0$ by the argument for $Q_1 \overline{m}^h$ with $\varphi \times \overline{m}^h \in L^4(\Omega ; L^4(0,T; {\mathbb {L}}^4))$, and the second term converges to 0 by (67). $\square$

We also obtain uniform bounds for $\overline{m}^h$ in weighted spaces.

Lemma 4.3

For any $w \ge 1$, the quadratic interpolation $\overline{m}^h$ satisfies

• (i) $\sup _h {\mathbb {E}}[ |\overline{m}^h|_{B_w}^2 ] < \infty$ for $B_w:= L^2(0,T; {\mathbb {L}}^2_w \cap \mathring{{\mathbb {H}}}^2) \cap {\mathcal {C}}([0,T]; {\mathbb {L}}^2_w \cap \mathring{{\mathbb {H}}}^1)$,

• (ii) $\sup _h {\mathbb {E}}[ |\overline{m}^h|^2_{W^{\alpha ,p}(0,T; {\mathbb {L}}^2_w)} ] < \infty$, for $p \in [2,\infty )$ and $\alpha \in (0, \frac{1}{2})$ such that $\alpha - \frac{1}{p} < \frac{1}{2}$,

• (iii) $|\overline{m}^h| \rightarrow 1$ in $L^2(\Omega ; L^2(0,T; {\mathbb {L}}^2))$.

Proof

$\underline{\text { Part (i).}}$ For every fixed $h > 0$, $\widehat{m}^h$ is in ${\mathcal {C}}([0,T]; {\mathbb {L}}^2_w \cap \mathring{{\mathbb {H}}}^1)$ and so does $\overline{m}^h$. Then part (i) follows directly from the estimates in (64) and (65).

$\underline{\text { Part (ii).}}$ Recall (59), we have from the definition (54) that

$$\begin{aligned} \overline{m}^h(t,x)&= I_0(x) + I_1(t,x) + I_2(t,x) + I_3(t,x) + I_4(t,x), \end{aligned}$$

where for $x \in [x_k,x_{k+1})$, $k \in {\mathbb {Z}}$,

$$\begin{aligned} I_0(x)&= \frac{1}{2} \left( m_0(x_k) + m_0(x_{k-1}) \right) + (x-x_k) \partial ^{h} m_0(x_{k-1}) + \frac{1}{2}(x-x_k)^2\Delta ^h m_0(x_k), \\ I_1(t,x)&= \frac{1}{2} \int _0^t \left( F^h(\widehat{m}^h(s,x_k)) + \frac{1}{2} S^h(\widehat{m}^h(s,x_k)) \right) \textrm{d}s \\&\quad + \frac{1}{2} \int _0^t \left( F^h(\widehat{m}^h(s,x_{k-1})) + \frac{1}{2} S^h(\widehat{m}^h(s,x_{k-1})) \right) \textrm{d}s, \\ I_2(t,x)&= (x-x_k) \int _0^t \partial ^{h} \left( F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right) (s,x_{k-1}) \ \textrm{d}s, \\ I_3(t,x)&= \frac{1}{2}(x-x_k)^2 \int _0^t \Delta ^h \left( F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right) (s,x_k) \ \textrm{d}s, \\ I_4(t,x)&= \frac{1}{2} \int _0^t \left( G^h(\widehat{m}^h(s,x_k)) + G^h(\widehat{m}^h(s,x_{k-1})) \right) \textrm{d}\widehat{W}^h(s) \\&\quad + (x-x_k) \int _0^t \partial ^{h} G^h(\widehat{m}^h(s,x_{k-1})) \ \textrm{d}\widehat{W}^h(s) \\&\quad + \frac{1}{2}(x-x_k)^2 \int _0^t \Delta ^h G^h(\widehat{m}^h(s,x_k)) \ \textrm{d}\widehat{W}^h(s). \end{aligned}$$

By the ${\mathbb {L}}^\infty$-estimate of $\overline{m}^h$ in (64),

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T |\overline{m}^h(t)|_{{\mathbb {L}}^2_w}^2 \ \textrm{d}t \right] \le 5^2 \pi T. \end{aligned}$$

For $I_1$, by Lemma 3.5, there exists a constant $a_1$ that may depend on $C_v, C_\kappa , \alpha , \gamma , T, K_{1,1}, K_{1,3}$ and $K_{2,1}$ such that

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T \left| F^h(\widehat{m}^h(t)) + \frac{1}{2} S^h(\widehat{m}^h(t)) \right| _{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \le a_1, \end{aligned}$$

For $I_2$ and $I_3$, since $|x-x_k| \le h$ and

$$\begin{aligned} h| \partial ^{h} u(x_{k-1}) |&\le |u(x_k)| + |u(x_{k-1})|, \\ h^2 |\Delta ^h u(x_k) |&\le |u(x_{k+1})| + |u(x_{k-1})| + 2|u(x_k)|, \end{aligned}$$

there also exist constants $a_2, a_3$ such that

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T \left| h \partial ^{h} \left( F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right) (t) \right| _{{\mathbb {L}}^2}^2 \textrm{d}t \right]&\le a_2, \\ {\mathbb {E}}\left[ \int _0^T \left| h^2 \Delta ^h \left( F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right) (t) \right| _{{\mathbb {L}}^2}^2 \textrm{d}t \right]&\le a_3. \end{aligned}$$

Similarly, for the stochastic integrals in $I_4$, we only need to verify that $\int _0^T G^h(\widehat{m}^h(s)) \textrm{d}\widehat{W}^h(s)$ is bounded in $L^p(\Omega ; W^{\alpha , p}(0,T; {\mathbb {L}}^2))$. By [17, Lemma 2.1], there exist a constant C depending on $\alpha , p, \gamma , T$ and a constant $a_4$ depending on $C, C_\kappa$ and $K_{1,p}$ such that for $p \in [2,\infty )$ and $\alpha \in (0, \frac{1}{2})$,

$$\begin{aligned}&{\mathbb {E}}\left[ \left| \int _0^T G^h(\widehat{m}^h(s)) \ \textrm{d}\widehat{W}^h(s) \right| ^p_{W^{\alpha ,p}(0,T; {\mathbb {L}}^2)} \right] \\&\quad = {\mathbb {E}}\left[ \left| \int _0^T G^h(m^h(s)) \ \textrm{d}W(s) \right| ^p_{W^{\alpha ,p}(0,T; {\mathbb {L}}_h^2)} \right] \\&\quad \le C \ {\mathbb {E}}\left[ \int _0^T \left( \sum _j q_j^2 \left| f_j G^h(m^h(s)) \right| ^2_{{\mathbb {L}}_h^2} \right) ^\frac{p}{2} \textrm{d}s \right] \\&\quad \le C |\kappa |_{{\mathbb {L}}^\infty }^p {\mathbb {E}}\left[ \int _0^T \left| G^h(\widehat{m}^h(s)) \right| ^p_{{\mathbb {L}}^2} \ \textrm{d}s \right] \le a_4. \end{aligned}$$

Since ${\mathbb {L}}^2 \hookrightarrow {\mathbb {L}}^2_w$ for $w \ge 1$, the estimates above hold for the ${\mathbb {L}}^2_w$-norm. By Lemma A.4, the embedding $W^{1,2}(0,T; {\mathbb {L}}^2_w) \hookrightarrow W^{\alpha ,p}(0,T; {\mathbb {L}}^2_w)$ is continuous for $\alpha - \frac{1}{p} < \frac{1}{2}$. Thus,

$$\begin{aligned} \sup _h {\mathbb {E}}\left[ |\overline{m}^h|_{W^{\alpha ,p}(0,T; {\mathbb {L}}^2_w)}^2 \right] < \infty . \end{aligned}$$

$\underline{\text { Part (iii).}}$ Since $|m^h(t,x)|=1$, ${\mathbb {P}}$-a.s. for all $(t,x) \in [0,T] \times {\mathbb {Z}}_h$, we observe that

$$\begin{aligned} |\overline{m}^h(t,x)| \le 1 + \left| \partial ^h m^h(t,x_{k-1})(x-x_k) + \frac{1}{2} \Delta ^h m^h(t,x_k)(x-x_k)^2 \right| , \end{aligned}$$

for $(t,x) \in [0,T] \times [x_k,x_{k+1})$, $k \in {\mathbb {Z}}$. This implies

$$\begin{aligned}&{\mathbb {E}}\left[ \int _0^T \int _{\mathbb {R}}\left| |\overline{m}^h(t,x)| -1 \right| ^2 \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}\left[ \int _0^T \sum _k \int _{x_k}^{x_{k+1}} \left| \partial ^{h} m^h(t,x_{k-1})(x-x_k) + \frac{1}{2} \Delta ^h m^h(t,x_k)(x-x_k)^2 \right| ^2 \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le \frac{2}{3} h^2 K_{1,1} T + \frac{1}{10} h^4 K_{3,1}, \end{aligned}$$

where the last inequality holds by Lemmata 3.5 and 3.7, and we obtain the convergence after taking $h \rightarrow 0$. $\square$

5 Existence of solution

In this section, we first show that the sequence $\{(\overline{m}^h, W)\}_h$ is tight and then by using the Skorohod theorem we obtain its almost sure convergence, up to a change of probability space. Finally, we prove that the limit is a solution of the stochastic LLS equation (4) in the sense of Definition 2.1.

5.1 Tightness and construction of new probability space and processes

Fix $w_1, w_2$ such that $w_2 > w_1 \ge 1$. Define

$$\begin{aligned} E_0&:= L^2(0,T; {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^2) \cap W^{\alpha , 4}(0,T; {\mathbb {L}}^2_{w_2}), \\ E&:= L^2(0,T; {\mathbb {H}}_{w_2}^1) \cap {\mathcal {C}}([0,T]; {\mathbb {H}}_{w_1}^{-1}). \end{aligned}$$

Recall ${\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^2 {\mathop {\hookrightarrow }\limits ^{\text {compact}}} {\mathbb {H}}^1_{w_2} \hookrightarrow {\mathbb {L}}^2_{w_2}$. By (99) and Lemma A.2,

$$\begin{aligned} E_0 \hookrightarrow L^2(0,T; {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^2) \cap W^{\beta ,2}(0,T; {\mathbb {L}}^2_{w_2}) {\mathop {\hookrightarrow }\limits ^{\text {compact}}} L^2(0,T; {\mathbb {H}}^1_{w_2}), \end{aligned}$$

where $\overline{m}^h \in E_0$, ${\mathbb {P}}$-a.s. by Lemma 4.3. Also, since the embeddings ${\mathbb {H}}^1_{w_1} \hookrightarrow {\mathbb {L}}^2_{w_2} \hookrightarrow {\mathbb {H}}_{w_1}^{-1}$ are compact and $4 \alpha >1$, it holds by Lemma A.3 that

$$\begin{aligned} W^{\alpha , 4}(0,T; {\mathbb {L}}^2_{w_2}) {\mathop {\hookrightarrow }\limits ^{\text {compact}}} {\mathcal {C}}([0,T]; {\mathbb {H}}^{-1}_{w_1}). \end{aligned}$$

In summary, $E_0$ is compactly embedded in E. For any $r>0$,

$$\begin{aligned} {\mathbb {P}}\left( |\overline{m}^h|_{E_0} > r \right) \le \frac{1}{r^2} {\mathbb {E}}\left[ |\overline{m}^h|^2_{E_0} \right] , \end{aligned}$$

where $\{ |\overline{m}^h|_{E_0} \le r \}$ is compact in E, and the right-hand-side converges to 0 as r tends to infinity. Therefore, the set of laws $\{ {\mathcal {L}}(\overline{m}^h) \}$ on the Banach space E is tight, which implies the following convergence result.

Lemma 5.1

There exists a probability space $(\Omega ^*, {\mathcal {F}}^*, {\mathbb {P}}^*)$ and there exists a sequence $(m_h^*, W_h^*)$ of $E \times {\mathcal {C}}([0,T]; H^2({\mathbb {R}}))$-valued random variables defined on $(\Omega ^*, {\mathcal {F}}^*, {\mathbb {P}}^*)$, such that the laws of $(\overline{m}^h, W)$ and $(m_h^*, W_h^*)$ on $E \times {\mathcal {C}}([0,T]; H^2({\mathbb {R}}))$ are equal for every h, and there exists an $E \times {\mathcal {C}}([0,T]; H^2({\mathbb {R}}))$-valued random variable $(m^*, W^*)$ defined on $(\Omega ^*, {\mathcal {F}}^*, {\mathbb {P}}^*)$ such that

$$\begin{aligned} m_h^* \rightarrow m^* \text { in } E, \quad {\mathbb {P}}^*\text {-a.s.} \end{aligned}$$

(68)

and

$$\begin{aligned} W_h^* \rightarrow W^* \text { in } {\mathcal {C}}([0,T]; H^2({\mathbb {R}})), \quad {\mathbb {P}}^*\text {-a.s.} \end{aligned}$$

(69)

Proof

Since $E \times {\mathcal {C}}([0,T]; H^2({\mathbb {R}}))$ is a separable metric space, the result holds by the Skorohod theorem. $\square$

Since the laws of $(\overline{m}^h, W)$ and $(m_h^*, W_h^*)$ on $E \times {\mathcal {C}}([0,T]; H^2({\mathbb {R}}))$ are equal, due to the following remark we obtain the same estimates for $m_h^*$.

Remark 5.2

By Kuratowski’s theorem, the Borel sets of

$$\begin{aligned} B := B_{w_1}= L^2(0,T; {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^2) \cap {\mathcal {C}}([0,T]; {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^1) \end{aligned}$$

are Borel sets of $E = L^2(0,T; {\mathbb {H}}^1_{w_2}) \cap {\mathcal {C}}([0,T]; {\mathbb {H}}^{-1}_{w_1})$ for $w_1 < w_2$, where

$$\begin{aligned} {\mathbb {P}}\left( \overline{m}^h \in B \right) = 1. \end{aligned}$$

We can assume that $m_h^*$ takes values in B and the laws on B of $\overline{m}^h$ and $m_h^*$ are equal.

By Remark 5.2, the sequence $(m_h^*)_h$ satisfies the same estimates as $(\overline{m}^h)_h$ on B. By (65), for any $p \in [1,\infty )$,

$$\begin{aligned} \sup _{h} {\mathbb {E}}^* \left[ \sup _{t \in [0,T]} |m_h^*(t)|_{{\mathbb {L}}^2_{w_1}}^{2p} \right]&< \infty , \end{aligned}$$

(70)

$$\begin{aligned} \sup _{h} {\mathbb {E}}^* \left[ \sup _{t \in [0,T]} |Dm_h^*(t)|_{{\mathbb {L}}^2}^{2p} \right]&< \infty , \end{aligned}$$

(71)

$$\begin{aligned} \sup _{h} {\mathbb {E}}^* \left[ \left( \int _0^T |D^2 m_h^*(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right) ^p \right]&< \infty . \end{aligned}$$

(72)

Since $|\rho _w'| \le w \rho$ for $w >0$, by Gagliardo–Nirenberg inequality,

$$\begin{aligned} |m_h^*(t) \rho _{w_1}^\frac{1}{2}|_{{\mathbb {L}}^\infty }&\le C\ |D(m_h^*(t) \rho _\frac{w_1}{2})|_{{\mathbb {L}}^2}^\frac{1}{2} \ |m_h^*(t) \rho _\frac{w_1}{2}|_{{\mathbb {L}}^2}^\frac{1}{2} \\&\le C \left( |Dm_h^*(t) \rho _\frac{w_1}{2} |_{{\mathbb {L}}^2} + |m_h^*(t) \rho _\frac{w_1}{2}^\prime |_{{\mathbb {L}}^2} \right) ^\frac{1}{2} \ |m_h^*(t)|_{{\mathbb {L}}^2_{w_1}}^\frac{1}{2} \\&\le C \left( |Dm_h^*(t)|_{{\mathbb {L}}^2} + \frac{w_1}{2} |m_h^*(t)|_{{\mathbb {L}}^2_{w_1}} \right) ^\frac{1}{2} \ |m_h^*(t)|_{{\mathbb {L}}^2_{w_1}}^\frac{1}{2} \\&\le \frac{C}{2} \left( |Dm_h^*(t)|_{{\mathbb {L}}^2} + \frac{w_1}{2} |m_h^*(t)|_{{\mathbb {L}}^2_{w_1}} + |m_h^*(t)|_{{\mathbb {L}}^2_{w_1}} \right) , \end{aligned}$$

which implies

$$\begin{aligned} \sup _{h} {\mathbb {E}}^* \left[ \sup _{t \in [0,T]} |m_h^*(t) \rho _{w_1}^\frac{1}{2}|_{{\mathbb {L}}^\infty }^{2p} \right] < \infty , \quad p \in [1, \infty ). \end{aligned}$$

(73)

Thus, by (70) – (73), for $p \in [1,\infty )$,

$$\begin{aligned} \begin{aligned} \sup _{h} {\mathbb {E}}^* \left[ \left( \int _0^T |m_h^*(t) \times Dm_h^*(t)|_{{\mathbb {L}}^2_{w_1}}^2 \ \textrm{d}t \right) ^p \right]&<\infty , \\ \sup _{h} {\mathbb {E}}^* \left[ \left( \int _0^T |m_h^*(t) \times D^2 m_h^*(t) |_{{\mathbb {L}}^2_{w_1}}^2 \ \textrm{d}t \right) ^p \right]&< \infty , \\ \sup _{h} {\mathbb {E}}^* \left[ \left( \int _0^T |m_h^*(t) \times (m_h^*(t) \times D^2 m_h^*(t)) |_{{\mathbb {L}}_{2w_1}^2}^2 \ \textrm{d}t \right) ^p \right]&< \infty , \end{aligned} \end{aligned}$$

(74)

As in (53),

$$\begin{aligned} |Dm_h^*(t)|_{{\mathbb {L}}^4}^4&\le \frac{C^2}{2} \ |D^2 m_h^*(t)|_{{\mathbb {L}}^2}^2 + \frac{1}{2} |Dm_h^*(t)|_{{\mathbb {L}}^2}^6, \end{aligned}$$

which implies

$$\begin{aligned} \sup _{h} {\mathbb {E}}^* \left[ \left( \int _0^T |D m_h^*(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right) ^p \right] < \infty , \quad p \in [1,\infty ). \end{aligned}$$

(75)

5.2 Identification of the limit $(m^*, W^*)$

5.2.1 Convergence of functions of $m_h^*$

For $p \in [1,\infty )$, by the pointwise convergence of $m_h^*$ in (68) and the uniform integrability of $m_h^*$ and $D m_h^*$ in (70) – (71), we have

$$\begin{aligned} m_h^* \rightarrow m^*&\quad \hbox { in}\ L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2})) , \end{aligned}$$

(76)

$$\begin{aligned} Dm_h^* \rightarrow Dm^*&\quad \hbox { in}\ L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2})) . \end{aligned}$$

(77)

By (71), $Dm_h^*$ also converges weakly to a measurable process X in $L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2))$, which implies that $X = Dm^* \in L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2))$ by the uniqueness of the limit of weak convergence in $L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$. By (77) and integration-by-parts,

$$\begin{aligned} D^2m_h^* \rightharpoonup D^2 m^* \quad \hbox { in}\ L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2})). \end{aligned}$$

(78)

Similarly, by (72), $D^2m_h^*$ converges weakly to a measurable process Y in $L^{2p}(\Omega ^*; {\mathbb {L}}^2(0,T; {\mathbb {L}}^2))$, thus $Y = D^2 m^* \in L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2))$ and

$$\begin{aligned} D^2m_h^* \rightharpoonup D^2 m^* \quad \hbox { in}\ L^{2p}(\Omega ^*; L^2(0,T; {\mathbb {L}}^2)). \end{aligned}$$

(79)

Lemma 5.3

We have

• (i) $|m^*(t,x)| = 1$, (t, x) -a.e. ${\mathbb {P}}^*$-a.s.

• (ii) $m_h^* \rho _{w_2}^\frac{1}{2} \rightarrow m^* \rho _{w_2}^\frac{1}{2}$ in $L^p(\Omega ^*; L^p(0,T; {\mathbb {L}}^p))$, for $p \in [2,\infty )$,

• (iii) $Dm^* \in L^4(\Omega ^*; L^4(0,T; {\mathbb {L}}^4)) \cap L^p(\Omega ^*; L^\infty (0,T;{\mathbb {L}}^2))$, for $p \in [2,\infty )$.

Proof

$\underline{\text { Part (i).}}$ Recall Lemma 4.3(iii); a similar argument holds for ${\mathbb {L}}^2_{w_2}$ (in place of ${\mathbb {L}}^2$). Then,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}\left| |m^*(t,x)| - 1 \right| ^2 \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le 2{\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}\left| |m_h^*(t,x)|-1 \right| ^2 \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] + 2{\mathbb {E}}^* \left[ \int _0^T |m_h^*(t) - m^*(t)|^2_{{\mathbb {L}}^2_{w_2}} \textrm{d}t \right] , \end{aligned}$$

where the first expectation on the right-hand side converges to 0 since the laws of $\overline{m}^h$ and $m_h^*$ are the same on $L^2(0,T;{\mathbb {L}}^2_{w_2})$, and the second expectation converges to 0 by (76). Thus,

$$\begin{aligned} {\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}\left| |m^*(t,x)| - 1 \right| ^2 \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] = 0, \end{aligned}$$

which implies $|m^*(t,x)| = 1$ a.e. on $[0,T] \times {\mathbb {R}}$, ${\mathbb {P}}^*$-a.s. This also means

$$\begin{aligned} m^* \in L^p(\Omega ^*; L^p(0,T; {\mathbb {L}}^p_w)), \quad \forall p \in [1,\infty ), \ w \ge 1. \end{aligned}$$

$\underline{\text { Part (ii).}}$ For $w_2 > w_1 \ge 1$ and $p \in [2,\infty )$,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2} |_{{\mathbb {L}}^p}^p \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}^* \left[ \sup _{t \in [0,T]} \left| (m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2} \right| _{{\mathbb {L}}^\infty }^{p-1} \int _0^T \int _{\mathbb {R}}|(m_h^*(t,x) - m^*(t,x)) \rho _{w_2}^\frac{1}{2}(x)| \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le C \left( {\mathbb {E}}^* \left[ \sup _{t \in [0,T]} \left( |m_h^*(t) \rho _{w_1}^\frac{1}{2}|_{{\mathbb {L}}^\infty }^{p-1} + |m^*(t) \rho _{w_1}^\frac{1}{2}|_{{\mathbb {L}}^\infty }^{p-1} \right) ^2 \right] \right) ^\frac{1}{2} \\&\qquad \times \left( {\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}|m_h^*(t,x) - m^*(t,x)|^2 \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

for some constant C that depends on p and T. Then, by the ${\mathbb {L}}^\infty$-estimate (73), Lemma 5.3(i), (5) and the strong convergence (76),

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^*\left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2} |_{{\mathbb {L}}^p}^p \ \textrm{d}t \right] = 0, \quad p \in [2, \infty ). \end{aligned}$$

$\underline{\text { Part (iii).}}$ From part (i), we have $\frac{1}{2} D|m^*(t,x)|^2 = \langle m^*, Dm^* \rangle (t,x) = 0$ and thus

$$\begin{aligned} \left\langle m^*(t,x), D^2m^*(t,x) \right\rangle = - |Dm^*(t,x)|^2, \end{aligned}$$

for (t, x) -a.e. ${\mathbb {P}}^*$-a.s. Since $D^2m^* \in L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2))$, we deduce

$$\begin{aligned} {\mathbb {E}}^*\left[ \int _0^T |Dm^*(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right]&= {\mathbb {E}}^*\left[ \int _0^T \int _{\mathbb {R}}\left\langle m^*(t,x), D^2m^*(t,x) \right\rangle ^2 \ \textrm{d}x \ \textrm{d}t \right] \\&\le {\mathbb {E}}^*\left[ \int _0^T |D^2 m^*(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] < \infty . \end{aligned}$$

As in [11], we extend the definition of the ${\mathbb {L}}^2_{w_2} \cap \mathring{{\mathbb {H}}}^1$-norm to ${\mathbb {H}}^{-1}_{w_1}$ such that $|u|_{{\mathbb {L}}^2_{w_2} \cap \mathring{{\mathbb {H}}}^1} = \infty$ if the function u is in ${\mathbb {H}}^{-1}_{w_1}$ but not ${\mathbb {L}}^2_{w_2} \cap \mathring{{\mathbb {H}}}^1$, where the extended map

$$\begin{aligned} u \mapsto \sup _{t \in [0,T]} |u(t)|_{{\mathbb {L}}^2_{w_2} \cap \mathring{{\mathbb {H}}}^1}, \quad u \in {\mathcal {C}}([0,T]; {\mathbb {H}}^{-1}_{w_1}), \end{aligned}$$

is lower semicontinuous. Then, by the pointwise convergence (68), Fatou’s lemma and (71),

$$\begin{aligned} {\mathbb {E}}^*\left[ \sup _{t \in [0,T]} |Dm^*(t)|_{{\mathbb {L}}^2}^p \right]&\le \liminf _{h \rightarrow 0} {\mathbb {E}}^*\left[ \sup _{t \in [0,T]} |m_h^*(t)|_{{\mathbb {L}}^2_{w_2} \cap \mathring{{\mathbb {H}}}^1}^p \right] <\infty , \end{aligned}$$

for $p \in [2,\infty )$. $\square$

Lemma 5.4

We have the following strong convergences:

• (i) $m_h^* \times Dm_h^* \rightarrow m^* \times Dm^*$ and $\langle m_h^*, Dm_h^* \rangle \rightarrow 0$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (ii) $m_h^* \times \left( m_h^* \times D m_h^* \right) \rightarrow m^* \times (m^* \times Dm^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}_{w_1+w_2}^2))$.

Proof

$\underline{\text { Part (i).}}$ Note that

$$\begin{aligned} m_h^* \times Dm_h^* - m^* \times Dm^* = \left( m_h^* - m^* \right) \times Dm_h^* + m^* \times \left( Dm_h^* - Dm^* \right) . \end{aligned}$$

Then by Hölder’s inequality,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \times Dm_h^*(t)|_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] \\&\le \left( {\mathbb {E}}^*\left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2} |_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}^*\left[ \int _0^T |Dm_h^*(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

where the last line converges to 0 by Lemma 5.3(ii) and (75). Similarly, by Lemma 5.3(i),

$$\begin{aligned} {\mathbb {E}}^* \left[ \int _0^T |m^*(t) \times (Dm_h^*(t) - Dm^*(t)) |_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right]&\quad \le {\mathbb {E}}^*\left[ \int _0^T |Dm_h^*(t) - Dm^*(t)|_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] , \end{aligned}$$

where the right-hand side converges to 0 by (77). Therefore,

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^* \left[ \int _0^T |m_h^*(t) \times Dm_h^*(t) - m^*(t) \times Dm^*(t) |_{{\mathbb {L}}^2_w}^2 \ \textrm{d}t \right] = 0. \end{aligned}$$

(80)

Since $|m^*(t,x)|=1$, we have $\langle m^*, Dm^* \rangle (t,x)=0$. By the same argument as above (replacing cross product with scalar product), $\langle m_h^*, Dm_h^* \rangle \rightarrow \langle m^*, Dm^* \rangle = 0$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$.

$\underline{\text { Part (ii).}}$ Note that

$$\begin{aligned}&m_h^* \times \left( m_h^* \times Dm_h^* \right) - m^* \times \left( m^* \times Dm^* \right) \\&\quad = \left( m_h^* - m^* \right) \times \left( m_h^* \times Dm_h^* \right) + m^* \times \left( m_h^* \times Dm_h^* - m^* \times Dm^* \right) . \end{aligned}$$

Then, with $\rho _{w_1 + w_2} = \rho _{w_1} \rho _{w_2}$,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left| (m_h^*(t) - m^*(t)) \times \left( m_h^*(t) \times Dm_h^*(t) \right) \right| _{{\mathbb {L}}^2_{w_1+w_2}}^2 \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2} |_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^{\frac{1}{2}} \left( {\mathbb {E}}^* \left[ \int _0^T | m_h^*(t) \rho _{w_1}^\frac{1}{2} \times Dm_h^*(t) |_{{\mathbb {L}}^4}^4 \right] \right) ^\frac{1}{2} \\&\quad \le \left( {\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2} |_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^{\frac{1}{2}} \\&\qquad \times \left( {\mathbb {E}}^*\left[ \sup _{t \in [0,T]} | m_h^*(t) \rho _{w_1}^\frac{1}{2} |_{{\mathbb {L}}^\infty }^8 \right] \right) ^{\frac{1}{4}} \left( {\mathbb {E}}^*\left[ \left( \int _0^T \left| Dm_h^*(t) \right| _{{\mathbb {L}}^4}^4 \textrm{d}t \right) ^2 \right] \right) ^{\frac{1}{4}}, \end{aligned}$$

where the first expectation in the last inequality converges to 0 by Lemma 5.3(ii) and the second and third expectations are finite by (73) and (75). Also,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left| m^*(t) \times \left( m_h^*(t) \times Dm_h^*(t) - m^*(t) \times Dm^*(t) \right) \right| _{{\mathbb {L}}^2_{w_1+w_2}}^2 \ \textrm{d}t \right] \end{aligned}$$

converges to 0 Lemma 5.3(i) and part (i). Then, the strong convergence of $m_h^* \times (m_h^* \times Dm_h^*)$ follows as desired. $\square$

Lemma 5.5

Assume that $w_2 \ge 4w_1$. For any measurable process $\varphi \in L^4(\Omega ^*; {\mathbb {L}}^4(0,T; {\mathbb {L}}^4_{w_2}))$, we have the following weak convergences (with test function $\varphi$):

• (i) $m_h^* \times (m_h^* \times D m_h^*) \rightharpoonup m^* \times (m^* \times Dm^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (ii) $|m_h^* \times Dm_h^*|^2 m_h^* \rightharpoonup |m^* \times Dm^*|^2m^*$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (iii) $Dm_h^* \times (m_h^* \times Dm_h^*) \rightharpoonup Dm^* \times (m^* \times Dm^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (iv) $\langle m_h^*, Dm_h^* \rangle m_h^* \times Dm_h^* \rightharpoonup 0$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (v) $m_h^* \times D^2m_h^* \rightharpoonup m^* \times D^2m^*$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (vi) $m_h^* \times (m_h^* \times D^2m_h^*) \rightharpoonup m^* \times (m^* \times D^2m^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$.

Proof

$\underline{\text { Part (i).}}$ As in lemma 5.4(ii), we first observe that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^*\left[ \int _0^T \left\langle (m_h^* - m^*) \times (m_h^* \times Dm_h^*), \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}}(t) \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \int _0^T | \left( m_h^*(t) - m^*(t) \right) \rho _{w_2}^\frac{1}{2} |_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T | m_h^*(t) \times Dm_h^*(t) \rho _{w_2}^\frac{1}{4} |_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2} \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \right] \right) ^\frac{1}{4}, \end{aligned} \end{aligned}$$

(81)

where the first expectation in the last line converges to 0 by Lemma 5.3(ii), the second and the third expectations are finite by (74) with $w_2 \ge 2w_1$ (equivalently, $\rho _{w_2} \le \rho _{w_1}^2$) and $\varphi \in L^4(\Omega ^*; L^4(0,T; {\mathbb {L}}^4_{w_2}))$. Since $|m^*(t,x)| = 1$ from Lemma 5.3(i), we have $m^* \times \varphi \in L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$. Then by Lemma 5.4(i),

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^*\left[ \int _0^T \left\langle m^* \times (m_h^* \times Dm_h^* - m^* \times Dm^*), \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}}(t) \ \textrm{d}t \right] = 0. \end{aligned}$$

(82)

Combining (81) and (82), we have the desired weak convergence for part (i).

$\underline{\text { Part (ii).}}$

$$\begin{aligned}&\left| m_h^* \times Dm_h^* \right| ^2m_h^* - |m^* \times Dm^*|^2m^* \\&\quad = \left( \left| m_h^* \times Dm_h^* \right| ^2- |m^* \times Dm^*|^2 \right) m_h^* + |m^* \times Dm^*|^2 \left( m_h^* - m^* \right) \\&\quad \le \left| m_h^* \times Dm_h^* - m^* \times Dm^* \right| \left| m_h^* \times Dm_h^* + m^* \times Dm^* \right| \ |m_h^*| + |m^* \times Dm^*|^2 \ |m_h^* - m^*|. \end{aligned}$$

Then, for the first term in the line above,

$$\begin{aligned} & {\mathbb {E}}^*\left[ \int _0^T \left\langle \left| m_h^* \times Dm_h^* - m^* \times Dm^* \right| \left| m_h^* \times Dm_h^* + m^* \times Dm^* \right| m_h^*, \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}}(t) \ \textrm{d}t \right] \nonumber \\ & \quad \le \left( {\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}\left( \left| m_h^* \times Dm_h^* \right| + |m^* \times Dm^*| \right) ^2 |m_h^*|^2 |\varphi |^2 (t,x) \ \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] \right) ^\frac{1}{2} \nonumber \\ & \qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T \left| m_h^* \times Dm_h^* - m^* \times Dm^*\right| _{{\mathbb {L}}^2_{w_2}}^2 (t) \ \textrm{d}t \right] \right) ^\frac{1}{2}. \end{aligned}$$

(83)

We show that the first expectation on the right-hand side of (83) is finite. Since $w_2 \ge 4w_1$, it holds that $\rho _{w_2} \le \rho _{w_1}^2 \rho _{w_2}^\frac{1}{2}$ and

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}\left| m_h^* \times Dm_h^* \right| ^2 |m_h^*|^2 |\varphi |^2 (t,x) \ \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}^*\left[ \sup _{t \in [0,T]} |m_h^*(t) \rho _{w_1}^\frac{1}{2} |_{{\mathbb {L}}^\infty }^4 \int _0^T \int _{\mathbb {R}}|Dm_h^*|^2 |\varphi \rho _{w_2}^\frac{1}{4}|^2 (t,x) \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \sup _{t \in [0,T]} |m_h^*(t) \rho _{w_1}^\frac{1}{2}|_{{\mathbb {L}}^\infty }^{16} \right] \right) ^\frac{1}{4} \times \left( {\mathbb {E}}^*\left[ \left( \int _0^T |Dm_h^*(t)|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right) ^2 \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

where three expectations in the last inequality are finite by (73), (75) and $\varphi \in L^4(\Omega ^*; L^4(0,T; {\mathbb {L}}^4_{w_2}))$. Similarly, by Lemma 5.3(i) and (iii),

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \int _{\mathbb {R}}\left| m^* \times Dm^* \right| ^2 |\varphi |^2(t,x) \ \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \int _0^T |Dm^*(t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} < \infty . \end{aligned}$$

Hence, the left-hand side of (83) converges to 0 as $h \rightarrow 0$ by Lemma 5.4(i). Similarly, with $|m^*(t,x)|=1$, ${\mathbb {P}}^*$-a.s. we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^*\left[ \int _0^T \int _{\mathbb {R}}\left\langle |m^* \times Dm^*|^2 \left( m_h^*- m^* \right) (t,x), \varphi (t,x) \right\rangle \rho _{w_2}(x) \ \textrm{d}x \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \int _0^T |Dm^*(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{2} \times \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4}\\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2}|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \end{aligned} \end{aligned}$$

(84)

where the last line converges to 0 by Lemma 5.3(ii) – (iii). Combining (83) and (84), we have the desired weak convergence for part (ii).

$\underline{\text { Part (iii).}}$ Note that

$$\begin{aligned}&Dm_h^* \times (m_h^* \times Dm_h^*) - Dm^* \times (m^* \times Dm^*) \\&\quad = (Dm_h^* - Dm^*) \times (m^* \times Dm^*) + Dm_h^* \times (m_h^* \times Dm_h^* - m^* \times Dm^*). \end{aligned}$$

Then,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left\langle (Dm_h^*(t) - Dm^*(t)) \times (m^*(t) \times Dm^*(t)), \varphi (t) \rho _{w_2} \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \int _0^T |Dm^*(t) \rho _{w_2}^\frac{1}{4}|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \times \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4}\\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T |Dm_h^*(t) - Dm^*(t)|_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

where the last line converges to 0 by Lemma 5.3(iii) and (77). Similarly,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left\langle Dm_h^*(t) \times (m_h^*(t) \times Dm_h^*(t) - m^*(t) \times Dm^*(t)), \varphi (t) \rho _{w_2} \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \int _0^T |Dm_h^*(t) \rho _{w_2}^\frac{1}{4}|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \times \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4}\\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T |m_h^*(t) \times Dm_h^*(t) - m^*(t) \times Dm^*(t)|_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

which converges to 0 by (75) and Lemma 5.4(i). Together, we have

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^* \left[ \int _0^T \left| \langle Dm_h^* \times (m_h^* \times Dm_h^*) - Dm^* \times (m^* \times Dm^*), \varphi \rangle _{{\mathbb {L}}^2_{w_2}}(t) \right| \textrm{d}t \right] = 0. \end{aligned}$$

$\underline{\text { Part (iv).}}$ Again, since $w_2 \ge 4 w_1$, we have $\rho _{w_2}^\frac{1}{4} \le \rho _{w_1}$. Then,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left\langle \ \langle m_h^*(t) , Dm_h^*(t) \rangle m_h^*(t) \times Dm_h^*(t), \varphi (t) \rho _{w_2} \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^*\left[ \int _0^T |m_h^*(t) \times Dm_h^*(t) \rho _{w_2}^\frac{1}{4}|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}^*\left[ \int _0^T |\langle m_h^*(t), Dm_h^*(t) \rangle |_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2} \\&\quad \le \left( {\mathbb {E}}^*\left[ \sup _{t \in [0,T]} |m_h^*(t) \rho _{w_1}|_{{\mathbb {L}}^\infty }^8 \right] \right) ^{\frac{1}{8}} \left( {\mathbb {E}}^* \left[ \left( \int _0^T |Dm_h^*(t)|^4_{{\mathbb {L}}^4} \ \textrm{d}t \right) ^2 \right] \right) ^\frac{1}{8} \\&\qquad \times \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}^*\left[ \int _0^T |\langle m_h^*(t), Dm_h^*(t) \rangle |_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2} \end{aligned}$$

where the right-hand side converges to 0 by (73) (with $\rho _{w_1}^\frac{1}{2} \le 1$), (75) and the convergence of the scalar product in Lemma 5.4(i).

$\underline{\text { Part (v).}}$

$$\begin{aligned}&\langle m_h^* \times D^2m_h^* - m^* \times D^2m^*, \varphi \rangle _{{\mathbb {L}}^2_{w_2}} \\&\quad = \langle (m_h^* - m^*) \times D^2 m_h^*, \varphi \rho _{w_2} \rangle _{{\mathbb {L}}^2} + \langle m^* \times (D^2m_h^* - D^2m^*), \varphi \rho _{w_2} \rangle _{{\mathbb {L}}^2}. \end{aligned}$$

Then, for the first term on the right-hand side,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left\langle (m_h^*(t) - m^*(t)) \times D^2 m_h^*(t), \varphi (t) \rho _{w_2} \right\rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2} \times \varphi (t)|_{{\mathbb {L}}^2} \ |D^2m_h^*(t)|_{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) -m^*(t)) \rho _{w_2}^\frac{1}{2}|_{{\mathbb {L}}^4} \ |\varphi (t) \rho _{w_2}^\frac{1}{2}|_{{\mathbb {L}}^4}\ |D^2m_h^*(t)|_{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2}|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \times \left( {\mathbb {E}}^* \left[ \int _0^T |\varphi (t) \rho _{w_2}^\frac{1}{2}|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}^* \left[ \int _0^T |D^2 m_h^*(t)|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

where the first expectation in the last inequality converges to 0 by Lemma 5.3(ii), the second expectation is finite as $\varphi \in L^4(\Omega ^*; {\mathbb {L}}^4(0,T; {\mathbb {L}}^4_{w_2}))$ and the final expectation is finite by (72). Thus, the left-hand side converges to 0 as $h \rightarrow 0$. Also, $m^* \times \varphi \in L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$ and then by the weak convergence (78),

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^* \left[ \int _0^T \langle m^*(t) \times (D^2m_h^*(t) - D^2m^*(t)), \varphi (t) \rangle _{{\mathbb {L}}^2_{w_2}} \ \textrm{d}t \right] = 0. \end{aligned}$$

Therefore,

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^* \left[ \int _0^T \left| \langle m_h^*(t) \times D^2 m_h^*(t) - m^*(t) \times D^2 m^*(t), \varphi (t) \rangle _{{\mathbb {L}}^2_{w_2}} \right| \textrm{d}t \right] = 0. \end{aligned}$$

$\underline{\text { Part (vi).}}$ Similarly,

$$\begin{aligned}&\left\langle m_h^* \times \left( m_h^* \times D^2m_h^* \right) - m^* \times \left( m^* \times D^2m^* \right) , \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}} \\&\quad = \langle (m_h^* - m^*) \times \left( m_h^* \times D^2 m_h^* \right) , \varphi \rangle _{{\mathbb {L}}^2_{w_2}} + \langle m^* \times (m_h^* \times D^2m_h^* - m^* \times D^2m^*), \varphi \rangle _{{\mathbb {L}}^2_{w_2}}. \end{aligned}$$

Then,

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \langle (m_h^*(t) - m^*(t)) \times \left( m_h^*(t) \times D^2 m_h^*(t) \right) , \varphi (t) \rangle _{{\mathbb {L}}^2_{w_2}} \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^* \left[ \int _0^T |(m_h^*(t) - m^*(t)) \rho _{w_2}^\frac{1}{2}|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \times \left( {\mathbb {E}}^* \left[ \int _0^T |\varphi (t) \rho _{w_2}^\frac{1}{4}|_{{\mathbb {L}}^4}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad \times \left( {\mathbb {E}}^* \left[ \int _0^T |m_h^*(t) \times D^2 m_h^*(t) \rho _{w_2}^\frac{1}{4}|_{{\mathbb {L}}^2}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2}, \end{aligned}$$

where the last line converges to 0 by Lemma 5.3(ii), $\varphi \in L^4(\Omega ^*; {\mathbb {L}}^4(0,T; {\mathbb {L}}^4_{w_2}))$ and (74) with $\rho _{w_1} \ge \rho _{w_2}^\frac{1}{4}$. Also, we have

$$\begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \langle m^*(t) \times (m_h^*(t) \times D^2m_h^*(t) - m^*(t) \times D^2m^*(t)), \varphi (t) \rho _{w_2} \rangle _{{\mathbb {L}}^2} \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}^* \left[ \int _0^T \langle m_h^*(t) \times D^2m_h^*(t) - m^*(t) \times D^2m^*(t), m^*(t) \times \varphi (t) \rangle _{{\mathbb {L}}^2_{w_2}} \ \textrm{d}t \right] , \end{aligned}$$

where $m^* \times \varphi \in L^4(\Omega ^*; L^4(0,T; {\mathbb {L}}^4_{w_2}))$ and thus by part (iii), the expectation in the last line above converges to 0. Therefore,

$$\begin{aligned} \lim _{h \rightarrow 0} {\mathbb {E}}^* \left[ \int _0^T \langle m_h^* \times \left( m_h^* \times D^2 m_h^* \right) - m^* \times \left( m^* \times D^2 m^* \right) , \varphi \rangle _{{\mathbb {L}}^2_{w_2}}(t) \ \textrm{d}t \right] = 0. \end{aligned}$$

$\square$

Lemma 5.6

Assume that $w_2 \ge 4w_1$. Recall the definitions (62), we have

• (i) $F_{\widehat{v}}(m_h^*) \rightharpoonup F(m^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (ii) $S_{\widehat{\kappa }}(m_h^*) \rightharpoonup S(m^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$,

• (iii) $\kappa G(m_h^*) \rightarrow \kappa G(m^*)$ (strongly) in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_1+w_2}))$.

Proof

As in Lemma 5.5, let $\varphi$ be an arbitrary measurable process in $L^4(\Omega ^*; L^4(0,T; {\mathbb {L}}^4_{w_2}))$. By (6) and (7), $\kappa ^2, \kappa \kappa ' \in {\mathbb {L}}^\infty \cap {\mathbb {H}}^1$ and $v \in {\mathcal {C}}([0,T]; {\mathbb {L}}^\infty \cap {\mathbb {H}}^1)$. Then for $y = \kappa ^2, (\kappa ^2)^-, \kappa \kappa ', v$, any piecewise constant approximation (in the x-variable) z of y satisfies

$$\begin{aligned} z \rightarrow y \quad \text { in } L^4(0,T; {\mathbb {L}}^4_{w_2}). \end{aligned}$$

(85)

For example, the approximation z can be taken to be $\widehat{y}^-$ or $\widehat{y}$. Let u be a function such that $u(m_h^*) \in L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$. Then,

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}^* \left[ \int _0^T \left\langle z(t) u(m_h^*(t)) - y(t) u(m^*(t)), \varphi (t) \right\rangle _{{\mathbb {L}}^2_{w_2}} \ \textrm{d}t \right] \\&\quad \le {\mathbb {E}}^*\left[ \int _0^T \left\langle (z - y) \times u(m_h^*), \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}}(t) \ \textrm{d}t \right] \\&\qquad + {\mathbb {E}}^*\left[ \int _0^T \left\langle y\left( u(m_h^*) - u(m^*) \right) , \varphi \right\rangle _{L^2_{w_2}}(t) \ \textrm{d}t \right] \\&\quad \le \left( {\mathbb {E}}^* \left[ \int _0^T |z(t) - y(t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \left( {\mathbb {E}}^*\left[ \int _0^T |u(m_h^*(t))|_{{\mathbb {L}}^2_{w_2}}^2 \ \textrm{d}t \right] \right) ^\frac{1}{2}\\&\qquad \left( {\mathbb {E}}^*\left[ \int _0^T |\varphi (t)|_{{\mathbb {L}}^4_{w_2}}^4 \ \textrm{d}t \right] \right) ^\frac{1}{4} \\&\qquad + {\mathbb {E}}^*\left[ \int _0^T \langle u(m_h^*(t)) - u(m^*(t)), y(t)\varphi (t) \rangle _{{\mathbb {L}}^2_{w_2}} \ \textrm{d}t \right] . \end{aligned} \end{aligned}$$

(86)

If $u(m_h^*) \rightharpoonup u(m^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}_{w_2}^2))$, then the right-hand side of (86) converges to 0 by (85) and $y \varphi \in L^4(\Omega ^*; L^4(0,T; {\mathbb {L}}^4_{w_2}))$.

$\underline{\text { Part (i).}}$ Let $u(m_h^*) = m_h^* \times (m_h^* \times Dm_h^*)$ and let $y = v$. The result follows immediately from (86) and Lemma 5.5(v) and (vi).

$\underline{\text { Part (ii).}}$ Since $w_2 \ge 4w_1$, we have $\rho _{w_2} \le \rho _{w_1}^4$, We observe that from (75) and (73) that

$$\begin{aligned} Dm_h^* \times (m_h^* \times Dm_h^*), \quad |m_h^* \times Dm_h^*|^2 m_h^*, \quad \langle m_h^*, Dm_h^* \rangle m_h^* \times Dm_h^* \end{aligned}$$

are in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}_{w_2}^2))$.

Taking the following choices of u, y and z:

$$\begin{aligned} u(m_h^*)&= (\gamma ^2-1) m_h^* \times (m_h^* \times D^2 m_h^*) - 2 \gamma m_h^* \times D^2 m_h^*, & \quad y = \kappa ^2, & \quad z = \frac{1}{2} \left( (\widehat{\kappa }^-)^2 + \widehat{\kappa }^2 \right) , \\ u(m_h^*)&= \gamma ^2 Dm_h^* \times (m_h^* \times Dm_h^*) + |m_h^* \times Dm_h^*|^2 m_h^*, & \quad y = \kappa ^2, & \quad z = \widehat{\kappa }^2, \\ u(m_h^*)&= 2 \gamma \langle m_h^*, Dm_h^* \rangle \ m_h^* \times Dm_h^*, & \quad y = \kappa ^2, & \quad z = (\widehat{\kappa }^2)^-, \\ u(m_h^*)&= \left[ (\gamma ^2-1) m_h^* \times (m_h^* \times Dm_h^*) - 2\gamma m_h^* \times Dm_h^* \right] , & \quad y = \kappa \kappa ', & \quad z = \widehat{\kappa \kappa '}, \end{aligned}$$

and using Lemma 5.5(ii) – (vi), we follow again the argument (86) to obtain weak convergence of $S_{\widehat{\kappa }}(m_h^*)$ to $S(m^*)$ in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2_{w_2}))$.

$\underline{\text { Part (iii).}}$ The result follows from (6) and Lemma 5.4. $\square$

5.2.2 Wiener process

Define a sequence of processes $\{ \overline{M}_h \}_{h > 0}$ on $(\Omega ,{\mathcal {F}}, {\mathbb {P}})$ by

$$\begin{aligned} \overline{M}_h(t)&:= \int _0^t \left( G(\overline{m}^h(s)) - (\gamma P_1 + P_2)\overline{m}^h(s) \right) \textrm{d}\widehat{W}^h(s). \end{aligned}$$

Recall the equation of $\overline{m}^h$, we have from (63) that

$$\begin{aligned} \overline{M}_h(t)&= \overline{m}^h(t) - \widehat{m}_0^h - \int _0^t \left( F_{\widehat{v}}(\overline{m}^h(s))+ \frac{1}{2} S_{\widehat{\kappa }}(\overline{m}^h(s)) \right) \ \textrm{d}s\\ &\quad - R_0\overline{m}^h(t) - \int _0^t R^h \overline{m}^h(s) \ \textrm{d}s, \end{aligned}$$

where the operator $R^h$ is given by

$$\begin{aligned} R^h u&:= -\widehat{v}(\gamma P_1 + P_2)u + Q_1u + \alpha Q_2u +\frac{1}{4} \left( (\widehat{\kappa }^2)^- + \widehat{\kappa }^2 \right) \left[ 2 \gamma Q_1u - (\gamma ^2 -1) Q_2u \right] \\&\quad + \frac{1}{2}\left( \widehat{\kappa }^2 \left( P_3 + \gamma ^2 P_4 \right) u - 2 \gamma (\widehat{\kappa }^2)^- P_5u \right) + \frac{1}{2} \widehat{\kappa \kappa '} \left[ 2\gamma P_1 - (\gamma ^2-1)P_2 \right] u. \end{aligned}$$

Similarly, define a sequence of processes $\{ M_h^* \}_{h > 0}$ on $(\Omega ^*, {\mathcal {F}}^*, {\mathbb {P}}^*)$ by

$$\begin{aligned} M_h^*(t)&:= m_h^*(t) - \widehat{m}_0^h - \int _0^t \left( F_{\widehat{v}}(m_h^*(s)) + \frac{1}{2} S_{\widehat{\kappa }}(m_h^*(s)) \right) \ \textrm{d}s \\&\quad - R_0 m_h^*(t) - \int _0^t R^h m_h^*(s) \ \textrm{d}s. \end{aligned}$$

Lemma 5.7

For each $t \in (0,T]$, we have the following weak convergence in $L^2(\Omega ^*; {\mathbb {H}}^{-1}_{w_1})$:

$$\begin{aligned} M_h^*(t) \rightharpoonup M^*(t) := m^*(t) - m_0 - \int _0^t \left( F(m^*(s))+ \frac{1}{2} S(m^*(s)) \right) \ \textrm{d}s. \end{aligned}$$

Proof

Recall that ${\mathbb {H}}_{w_1}^1$ is compactly embedded in ${\mathbb {L}}^2_{w_2}$. Let $t \in (0,T]$ and $\varphi \in L^2(\Omega ^*; {\mathbb {H}}_{w_1}^1)$. By Remark 5.2, the two sets of remainders

$$\begin{aligned}&\{ R_0 \overline{m}^h, R_1 \overline{m}^h, P_1\overline{m}^h, \ldots , P_5\overline{m}^h, Q_1\overline{m}^h, Q_2\overline{m}^h \}, \\&\{ R_0 m_h^*, R_1 m_h^*, P_1m_h^*, \ldots , P_5m_h^*, Q_1m_h^*, Q_2m_h^* \}, \end{aligned}$$

have the same laws for $\overline{m}^h, m_h^* \in L^2(0,T; {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^2) \cap {\mathcal {C}}([0,T]; {\mathbb {L}}^2_{w_1} \cap \mathring{{\mathbb {H}}}^1)$. Then, by Lemma 4.2,

$$\begin{aligned}&\lim _{h \rightarrow 0} {\mathbb {E}}^* \left[ _{{\mathbb {H}}^{-1}_{w_1}} \left\langle R_0m_h^*(t) +\int _0^t R^hm_h^*(s) \ \textrm{d}s, \varphi \right\rangle _{{\mathbb {H}}^1_{w_1}} \right] \\&\quad = \lim _{h \rightarrow 0} {\mathbb {E}}^*\left[ \left\langle R_0m_h^*(t), \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}} + \int _0^t \left\langle R^h m_h^*(s), \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}} \ \textrm{d}s \right] = 0. \end{aligned}$$

By the pointwise convergence (68) of $m_h^*$ in ${\mathcal {C}}([0,T]; {\mathbb {H}}^{-1}_{w_1})$, Lemma 5.6(i) – (ii) and the convergence of piecewise constant approximations $\widehat{m}_0^h \rightarrow m_0$ in ${\mathbb {L}}^2_w$ for $w>w_1$ (due to $\partial _x m_0 \in {\mathbb {L}}^2$), we have

$$\begin{aligned}&\lim _{h \rightarrow 0} {\mathbb {E}}^*\left[ _{{\mathbb {H}}^{-1}_{w_1}} \left\langle M_h^*(t), \varphi \right\rangle _{{\mathbb {H}}^1_{w_1}} \right] \\&\quad = \lim _{h \rightarrow 0} {\mathbb {E}}^*\left[ _{{\mathbb {H}}^{-1}_{w_1}} \left\langle m_h^*(t)-\widehat{m}_0^h, \varphi \right\rangle _{{\mathbb {H}}^1_{w_1}} - \int _0^t \left\langle F_{\widehat{v}}(m_h^*(s)) + \frac{1}{2} S_{\widehat{\kappa }}(m_h^*(s)), \varphi \right\rangle _{{\mathbb {L}}^2_{w_2}} \textrm{d}s \right] \\&\qquad - \lim _{h \rightarrow 0} {\mathbb {E}}^*\left[ _{{\mathbb {H}}^{-1}_{w_1}} \left\langle R_0 m_h^*(t) + \int _0^t R^h m_h^*(s), \varphi \right\rangle _{{\mathbb {H}}^1_{w_1}} \right] \\&\quad = {\mathbb {E}}^*\left[ _{{\mathbb {H}}^{-1}_{w_1}} \left\langle M^*(t), \varphi \right\rangle _{{\mathbb {H}}^1_{w_1}} \right] . \end{aligned}$$

$\square$

Lemma 5.8

The process $W^*$ is a Q-Wiener process on $(\Omega ^*, {\mathcal {F}}^*, {\mathbb {P}}^*)$, and $W^*(t) - W^*(s)$ is independent of the $\sigma$-algebra generated by $m^*(r)$ and $W^*(r)$ for $r \in [0,s]$.

Proof

See [11, Lemma 5.2(i)] (using Lemma 5.1). $\square$

Lemma 5.9

For each $t \in [0,T]$,

$$\begin{aligned} M^*(t) = \int _0^t G(m^*(s)) \ \textrm{d}W^*(s). \end{aligned}$$

Proof

Fix h and $t \in (0,T]$. For each $n \in {\mathbb N}$, define the partition $\{ s_i^n = \frac{iT}{n}: i = 0, \ldots , n \}$. Define

$$\begin{aligned} \delta \widehat{W}^h(t,s_i^n)&:= \widehat{W}^h(t \wedge s_{i+1}^n) - \widehat{W}^h(t \wedge s_i^n), \\ \delta \widehat{W}_h^*(t,s_i^n)&:= \widehat{W}_h^*(t \wedge s_{i+1}^n) - \widehat{W}_h^*(t \wedge s_i^n), \\ \delta W^*(t,s_i^n)&:= W^*(t \wedge s_{i+1}^n) - W^*(t \wedge s_i^n), \end{aligned}$$

where $\widehat{W}_h^*(s)$ is the piecewise constant approximation of $W_h^*(s)$ (as in (56)) for every $s \in [0,T]$. As in (69), we also have

$$\begin{aligned} \widehat{W}_h^* \rightarrow W^* \text { in } {\mathcal {C}}([0,T]; L^2({\mathbb {R}})), \quad {\mathbb {P}}^*\text {-a.s.} \end{aligned}$$

(87)

Consider the following two ${\mathbb {L}}^2_{w_2}$-valued random variables:

$$\begin{aligned} \overline{Y}_{h,n}(t)&:= \overline{M}_h(t) - \sum _{i=0}^{n-1} \left( G(\overline{m}^h(s_i^n)) - \left( \gamma P_1 + P_2 \right) \overline{m}^h(s_i^n) \right) \delta \widehat{W}^h(t,s_i^n), \\ Y^*_{h,n}(t)&:= M_h^*(t) - \sum _{i=0}^{n-1} \left( G(m_h^*(s_i^n)) - \left( \gamma P_1 + P_2 \right) m_h^*(s_i^n) \right) \delta \widehat{W}_h^*(t,s_i^n). \end{aligned}$$

Following Remark 5.2, $\overline{Y}_{h,n}$ and $Y^*_{h,n}$ have the same distribution. As $n \rightarrow \infty$,

$$\begin{aligned} \overline{Y}_{h,n}(t) \rightarrow \overline{M}_h(t) - \int _0^t \left( G(\overline{m}^h(s)) - \left( \gamma P_1 + P_2 \right) \overline{m}^h(s) \right) \ \textrm{d}\widehat{W}^h(s) = 0 \text { in } L^2(\Omega ; {\mathbb {L}}^2_{w_2}). \end{aligned}$$

This implies that $Y^*_{h,n}(t)$ also converges to 0 in $L^2(\Omega ^*; {\mathbb {L}}^2_{w_2})$ as $n \rightarrow \infty$. Thus,

$$\begin{aligned} M_h^*(t) = \int _0^t \left( G(m_h^*(s)) - \left( \gamma P_1 + P_2 \right) m_h^*(s) \right) \ \textrm{d}\widehat{W}_h^*(s), \quad {\mathbb {P}}^*\text {-a.s.} \end{aligned}$$

We observe that

$$\begin{aligned}&{\mathbb {E}}^*\left[ M_h^*(t) - \int _0^t G(m^*(s)) \ \textrm{d}W^*(s) \right] \\&\quad = {\mathbb {E}}^*\left[ \int _0^t \left( G(m_h^*(s)) - (\gamma P_1 + P_2)m_h^*(s) - \sum _{i=0}^{n-1} G(m_h^*(s_i^n)) \mathbbm {1}_{(s_i^n, s_{i+1}^n]}(s) \right) \ \textrm{d}\widehat{W}_h^*(s) \right] \\&\qquad + {\mathbb {E}}^*\left[ \sum _{i=0}^{n-1} G(m_h^*(s_i^n)) \ \delta \widehat{W}_h^*(t,s_i^n)- \sum _{i=0}^{n-1} G(m^*(s_i^n)) \ \delta W^*(t \wedge s_i^n) \right] \\&\qquad + {\mathbb {E}}^*\left[ \int _0^t \left( \sum _{i=0}^{n-1} G(m^*(s_i^n)) \mathbbm {1}_{(s_i^n,s_{i+1}^n]}(s) - G(m^*(s)) \right) \ \textrm{d}W^*(s) \right] \\&\quad = {\mathbb {E}}^*[J_0^h] + {\mathbb {E}}^*[J_1^h] + {\mathbb {E}}^*[J_2]. \end{aligned}$$

For $J_0^h$ and $J_2$, let $\varepsilon > 0$ and choose $n \in {\mathbb N}$ such that

$$\begin{aligned} \left( {\mathbb {E}}^*\left[ \int _0^t \left| G(m^*(s)) - \sum _{i=0}^{n-1}G(m^*(s_i^n)) \mathbbm {1}_{(s_i^n, s_{i+1}^n]}(s) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \ \textrm{d}s \right] \right) ^\frac{1}{2} < \frac{\varepsilon }{2}. \end{aligned}$$

(88)

Since $\widehat{W}_h^*$ and $\widehat{W}^h$ have the same laws on ${\mathcal {C}}([0,T]; H^2({\mathbb {R}}))$, we have

$$\begin{aligned}&\left( {\mathbb {E}}^*\left[ |J_0^h|_{{\mathbb {H}}^{-1}_{w_1}}^2 \right] \right) ^\frac{1}{2} \\&\quad \le \left( {\mathbb {E}}^* \left[ \int _0^t \sum _j q_j^2 \left| \widehat{f}_j G(m_h^*(s)) - \widehat{f}_j G(m^*(s)) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \textrm{d}s \right] \right) ^\frac{1}{2} \\&\qquad + \left( {\mathbb {E}}^*\left[ \int _0^t \sum _j q_j^2 \left| \widehat{f}_j \left( G(m^*(s)) - \sum _{i=0}^{n-1}G(m^*(s_i^n)) \mathbbm {1}_{(s_i^n, s_{i+1}^n]}(s) \right) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \textrm{d}s \right] \right) ^\frac{1}{2} \\&\qquad + \left( {\mathbb {E}}^*\left[ \int _0^t \sum _j q_j^2 \left| \sum _{i=0}^{n-1} \left( \widehat{f}_j G(m^*(s_i^n)) - \widehat{f}_j G(m_h^*(s_i^n)) \right) \mathbbm {1}_{(s_i^n, s_{i+1}^n]}(s) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \textrm{d}s \right] \right) ^\frac{1}{2} \\&\qquad + \left( {\mathbb {E}}^*\left[ \int _0^t \sum _j q_j^2 \left| \widehat{f}_j (\gamma P_1 + P_2) m_h^*(s) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \ \textrm{d}s \right] \right) ^\frac{1}{2}. \end{aligned}$$

Recall that ${\mathbb {L}}^2_w \hookrightarrow {\mathbb {H}}^{-1}_{w_1}$ for all $w > w_1$. Let $w = w_1+w_2$. As $h \rightarrow 0$, the first and the third term on the right-hand side converges to 0 by Lemma 5.6(iii), the fourth term converges to 0 by Lemma 4.2 and (6), and the second term is less than $\frac{\varepsilon }{2}$ by (88). Hence, for a sufficiently small h, we have

$$\begin{aligned} |J_0^h|_{L^2(\Omega ^*; {\mathbb {H}}_{w_1}^{-1})} < \frac{\varepsilon }{2}. \end{aligned}$$

Similarly, $|J_2|_{L^2(\Omega ^*; {\mathbb {H}}_{w_1}^{-1})} < \frac{\varepsilon }{2}$.

For $J_1^h$, we have

$$\begin{aligned} {\mathbb {E}}^*\left[ |J_1^h|_{{\mathbb {H}}^{-1}_{w_1}}^2 \right]&\le {\mathbb {E}}^*\left[ \left| \sum _{i=0}^{n-1} \left( G(m_h^*(s_i^n)) - G(m^*(s_i^n)) \right) \delta W^*(t,s_i^n)\right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \right] \\&\quad + {\mathbb {E}}^*\left[ \left| \sum _{i=0}^{n-1} G(m_h^*(s_i^n)) \left( \delta \widehat{W}_h^*(t,s_i^n)- \delta W^*(t,s_i^n) \right) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \right] . \end{aligned}$$

Since $W^*$ is a Q-Wiener process, the first term on the right-hand side converges to 0 by Lemma 5.6(iii), Also, the second term converges to 0 by the pointwise convergence (69) (or (87)) and the result $G(m_h^*) \rho _{\frac{w}{2}} \in L^2(\Omega ^*; L^2(0,T;{\mathbb {L}}^\infty ))$, which can be deduced from the estimates (71), (72) and (73).

Therefore, for any sufficiently small h,

$$\begin{aligned} {\mathbb {E}}^*\left[ \left| M_h^*(t) - \int _0^t G(m^*(s)) \ \textrm{d}W^*(s) \right| _{{\mathbb {H}}^{-1}_{w_1}}^2 \right] < \varepsilon ^2. \end{aligned}$$

Using Lemma 5.7 and the uniqueness of weak limit, the proof is concluded. $\square$

We are ready to prove the main theorem.

5.2.3 Proof of Theorem 2.5

By Lemmata 5.7 and 5.9, $m^*$ satisfies the (12) in ${\mathbb {H}}^{-1}_{w_1}$. Moreover, using Lemma 5.3(i), we can simplify F, S and G:

$$\begin{aligned} F(m^*)&= -v \left( Dm^*+ \gamma m^* \times Dm^*\right) - m^* \times D^2m^* + \alpha D^2m^* + \alpha |Dm^*|^2m^*, \\ S(m^*)&= \kappa ^2 \left( (1-\gamma ^2) D^2 m^* - 2\gamma ^2 |Dm^*|^2 m^* -2 \gamma m^* \times D^2m^* \right) \\&\quad +\kappa \kappa ' \left( (1-\gamma ^2) Dm^* - 2\gamma m^* \times Dm^* \right) , \\ G(m^*)&= -Dm^* + \gamma m^* \times Dm^*, \end{aligned}$$

and each of them is in $L^2(\Omega ^*; L^2(0,T; {\mathbb {L}}^2))$, hence the equality (12) holds in ${\mathbb {L}}^2$. Recall the properties of $m^*$ shown previously in (79) and Lemma 5.3, we have now verified that $m^*$ is a solution of (8) in the sense of Definition 2.1. It only remains to show that $m-m_0 \in {\mathcal {C}}^\alpha ([0,T]; {\mathbb {L}}^2)$. For $s,t \in [0,T]$ and $p \in [1,\infty )$, there exists a constant C that may depend on $p,T,C_{\kappa }$ such that

$$\begin{aligned}&{\mathbb {E}}^* \left[ |m^*(t)-m^*(s)|_{{\mathbb {L}}^2}^{2p} \right] \\&\le |t-s|^p {\mathbb {E}}^* \left[ \left( \int _s^t \left| F(m^*(r)) + \frac{1}{2}S(m^*(r)) \right| _{{\mathbb {L}}^2}^2 \ dr \right) ^p \right] \\&\quad + {\mathbb {E}}^* \left[ \left( \int _s^t \sum _j q_j^2 |f_j G(m^*)(r)|^2_{{\mathbb {L}}^2} \ dr \right) ^p \right] \\&\le C |t-s|^p {\mathbb {E}}^* \left[ \sup _{r \in [0,T]} |Dm^*(r)|^{2p}_{{\mathbb {L}}^2} + \left( \int _s^t \left( |Dm^*(r)|^4_{{\mathbb {L}}^4} + |D^2m^*(r)|^2_{{\mathbb {L}}^2}\right) \ dr \right) ^p \right] , \end{aligned}$$

where the expectation on the right-hand side is finite. Then by Kolmogorov’s continuity criterion, $m^*(t)-m_0 \in {\mathcal {C}}^\alpha ([0,T]; {\mathbb {L}}^2)$, ${\mathbb {P}}^*$-a.s. for $\alpha \in (0,\frac{1}{2})$.

5.2.4 Proof of Theorem 2.6

Let $(m_1,W)$ and $(m_2,W)$ on $(\Omega ,{\mathcal {F}},\left( {\mathcal {F}}_t\right) ,{\mathbb {P}})$ be two solutions of (8) in the sense of Definition 2.1. Let $u = m_1 - m_2$ and $w>0$. Applying Itô’s lemma to $\frac{1}{2}|u(t)|_{{\mathbb {L}}^2_w}^2$,

$$\begin{aligned} \begin{aligned} \frac{1}{2} |u(t)|^2_{{\mathbb {L}}^2_w}&= |u(0)|^2_{{\mathbb {L}}^2_w} + \int _0^t \left\langle u(s), F(m_1(s)) - F(m_2(s)) \right\rangle _{{\mathbb {L}}^2_w} \ \textrm{d}t \\&\quad + \frac{1}{2} \int _0^t \left\langle u(s), S(m_1(s))-S(m_2(s)) \right\rangle _{{\mathbb {L}}^2_w} \ \textrm{d}t \\&\quad + \frac{1}{2} \int _0^t \sum _j q_j^2 \left| f_j (G(m_1(s)) - G(m_2(s))) \right| _{{\mathbb {L}}^2_w}^2 \ \textrm{d}t \\&\quad + \int _0^t \left\langle u(s), \left( G(m_1(s)) - G(m_2(s)) \right) \ \textrm{d}W(s) \right\rangle _{{\mathbb {L}}^2_w} \\&= |u(0)|^2_{{\mathbb {L}}^2_w}+ \int _0^t \left[ U_1(s) + U_2(s) + U_3(s) \right] \ \textrm{d}s + U_4(t), \end{aligned} \end{aligned}$$

(89)

$\underline{\text { An estimate on } \,\, U_1:}$

$$\begin{aligned} U_1(s)&= \left\langle u, F(m_1) - F(m_2) \right\rangle _{{\mathbb {L}}^2_w}(s) \\&= \left\langle u, v \left( -Du + \gamma u \times Dm_1 + \gamma m_2 \times Du \right) - u \times D^2m_1 - m_2 \times D^2 u \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad + \alpha \left\langle u, \left\langle D(m_1+m_2), Du \right\rangle m_2 + D^2 u + |Dm_1|^2 u \right\rangle _{{\mathbb {L}}^2_w}(s) \\&= \left\langle u, v \left( -Du + \gamma m_2 \times Du \right) \right\rangle _{{\mathbb {L}}^2_w}(s) + \left\langle u, -m_2 \times D^2 u \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad + \alpha \left\langle u, \left\langle D(m_1+m_2), Du \right\rangle m_2 \right\rangle _{{\mathbb {L}}^2_w}(s) + \alpha \left\langle D^2u, u \right\rangle _{{\mathbb {L}}^2_w}(s) + \alpha \left\langle u, |Dm_1|^2 u \right\rangle _{{\mathbb {L}}^2_w}(s). \end{aligned}$$

Then, for an arbitrary $\varepsilon >0$,

$$\begin{aligned} \begin{aligned} \left\langle u, v \left( -Du + \gamma m_2 \times Du \right) \right\rangle _{{\mathbb {L}}^2_w}(s)&= \left\langle u \rho _w^\frac{1}{2}, v(-Du + \gamma m_2 \times Du) \rho _w^\frac{1}{2} \right\rangle _{{\mathbb {L}}^2}(s) \\&\le \frac{1}{2\varepsilon ^2} C_v^2 (1 + \gamma ^2) |u(s)|_{{\mathbb {L}}^2_w}^2 + \varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2, \end{aligned} \end{aligned}$$

(90)

and

$$\begin{aligned} \begin{aligned} \left\langle u, -m_2 \times D^2u \right\rangle _{{\mathbb {L}}^2_w}(s)&= - \left\langle D^2u, u \times m_2 \rho _w \right\rangle _{{\mathbb {L}}^2}(s) \\&= \left\langle Du, Du \times m_2 \rho _w + u \times D(m_2 \rho _w) \right\rangle _{{\mathbb {L}}^2}(s) \\&= \left\langle Du, u \times Dm_2 \right\rangle _{{\mathbb {L}}^2_w} + \left\langle Du, u \times m_2 \rho _w' \rho _w^{-1} \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\le \frac{1}{2 \varepsilon ^2} \left( |Dm_2(s)|_{{\mathbb {L}}^\infty }^2+w^2 \right) |u(s)|_{{\mathbb {L}}^2_w}^2 + \varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2. \end{aligned} \end{aligned}$$

(91)

Similarly,

$$\begin{aligned} \alpha \left\langle u, \left\langle D(m_1+m_2), Du \right\rangle m_2 \right\rangle _{{\mathbb {L}}^2_w}(s) & \le \frac{1}{2\varepsilon ^2}\alpha ^2 |Dm_1(s) + Dm_2(s)|_{{\mathbb {L}}^\infty }^2 |u(s)|_{{\mathbb {L}}^2_w}^2 \nonumber \\ & \quad + \frac{1}{2} \varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2, \end{aligned}$$

(92)

and

$$\begin{aligned} \begin{aligned} \alpha \left\langle u, D^2u \right\rangle _{{\mathbb {L}}^2_w}(s)&= \alpha \left\langle u \rho _w, D^2u \right\rangle _{{\mathbb {L}}^2}(s) \\&= -\alpha \left\langle Du, D(u \rho _w) \right\rangle _{{\mathbb {L}}^2}(s) \\&= -\alpha |Du(s)|_{{\mathbb {L}}^2_w}^2 - \alpha \left\langle Du, u \rho _w' \rho _w^{-1} \right\rangle _{{\mathbb {L}}^2_w}(s) \\&= -\alpha |Du(s)|_{{\mathbb {L}}^2_w}^2 + \frac{1}{2} \varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2 + \frac{1}{2 \varepsilon ^2} \alpha ^2 w^2 |u|_{{\mathbb {L}}^2_w}^2. \end{aligned} \end{aligned}$$

(93)

Also,

$$\begin{aligned} \alpha \left\langle u, |Dm_1|^2u \right\rangle _{{\mathbb {L}}^2_w}(s) \le \alpha |Dm_1(s)|_{{\mathbb {L}}^\infty }^2 |u(s)|_{{\mathbb {L}}^2_w}^2. \end{aligned}$$

(94)

Hence,

$$\begin{aligned} U_1(s)&\le \psi _1(s) |u(s)|_{{\mathbb {L}}^2_w}^2 + (3 \varepsilon ^2 - \alpha ) |Du(s)|_{{\mathbb {L}}^2_w}^2, \end{aligned}$$

for the process $\psi _1$ given by

$$\begin{aligned} \begin{aligned} \psi _1(s)&= \frac{1}{2\varepsilon ^2} \left( C_v^2(1+\gamma ^2) + |Dm_2(s)|_{{\mathbb {L}}^\infty }^2 + w^2 + \alpha ^2 |Dm_1(s) + Dm_2(s)|_{{\mathbb {L}}^\infty }^2 + \alpha ^2 w^2 \right) \\&\quad + \alpha |Dm_1(s)|_{{\mathbb {L}}^\infty }^2. \end{aligned} \end{aligned}$$

(95)

For $i=1,2$, there exists a constant $C>0$ such that

$$\begin{aligned} \begin{aligned} {\mathbb {E}}^*\left[ \int _0^T |Dm_i|^2_{{\mathbb {L}}^\infty }(t) \ \textrm{d}t \right]&\le {\mathbb {E}}^*\left[ \int _0^T |Dm_i|^2_{{\mathbb {H}}^1}(t) \ \textrm{d}t \right] < \infty , \end{aligned} \end{aligned}$$

(96)

which implies $\int _0^T \psi _1(t) \ \textrm{d}t < \infty$, ${\mathbb {P}}$-a.s.

$\underline{\text { An estimate on } \,\, U_2: }$

$$\begin{aligned} U_2(s)&= \frac{1}{2} \left\langle u, S(m_1) - S(m_2) \right\rangle _{{\mathbb {L}}^2_w}(s) \\&= \frac{1}{2} \left\langle u, \kappa ^2 \left[ (1-\gamma ^2) D^2u - 2\gamma (u \times D^2 m_1 + m_2 \times D^2 u) \right] \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad - \gamma ^2 \left\langle u, \kappa ^2 \left[ \left\langle D(m_1+m_2), Du \right\rangle m_2 + |Dm_1|^2 u \right] \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad + \frac{1}{2} \left\langle u, \kappa \kappa ' \left[ (1-\gamma ^2) Du - 2 \gamma (u \times Dm_1 + m_2 \times Du) \right] \right\rangle _{{\mathbb {L}}^2_w}(s) \\&= \frac{1}{2} \left\langle u, \kappa \kappa ' \left[ (1-\gamma ^2) Du - 2 \gamma m_2 \times Du \right] \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad + \frac{1}{2} (1-\gamma ^2) \left\langle u, \kappa ^2 D^2 u \right\rangle _{{\mathbb {L}}^2_w}(s) - \gamma \left\langle u, \kappa ^2 m_2 \times D^2 u \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad - \gamma ^2 \left\langle u, \kappa ^2 \left\langle D(m_1+m_2), Du \right\rangle m_2 \right\rangle _{{\mathbb {L}}^2_w}(s) - \gamma ^2 \left\langle u, \kappa ^2 |Dm_1|^2 u \right\rangle _{{\mathbb {L}}^2_w}(s). \end{aligned}$$

Again, for $\varepsilon >0$,

$$\begin{aligned}&\frac{1}{2}\left\langle u, \kappa \kappa ' \left[ (1-\gamma ^2) Du - 2 \gamma m_2 \times Du \right] \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad \le \frac{1}{4 \varepsilon ^2} C_\kappa ^4 \left( (1-\gamma ^2)^2 + 4 \gamma ^2 \right) |u(s)|_{{\mathbb {L}}^2_w}^2 + \frac{1}{2}\varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2. \end{aligned}$$

As in (91) and (92),

$$\begin{aligned}&-\gamma \left\langle u, \kappa ^2 m_2 \times D^2 u \right\rangle _{{\mathbb {L}}^2_w}(s)\\&\quad = -\gamma \left\langle D^2u, \kappa ^2 u \times m_2 \rho _w \right\rangle _{{\mathbb {L}}^2}(s) \\&\quad = \gamma \left\langle Du, Du \times m_2 \kappa ^2 \rho _w + u \times D(m_2 \kappa ^2 \rho _w) \right\rangle _{{\mathbb {L}}^2}(s) \\&\quad = \gamma \left\langle Du, \kappa ^2 u \times Dm_2 + \kappa ^2 u \times m_2 \rho _w' \rho _w^{-1} + 2 \kappa \kappa ' u \times m_2 \right\rangle _{{\mathbb {L}}^2_w}(s) \\&\quad \le \frac{1}{2 \varepsilon ^2} \gamma ^2 C_\kappa ^4 \left( |Dm_2(s)|_{{\mathbb {L}}^\infty }^2 + w^2 + 4 \right) |u(s)|_{{\mathbb {L}}^2_w}^2 + \varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2, \end{aligned}$$

and

$$\begin{aligned} - \gamma ^2 \left\langle u, \kappa ^2 \left\langle D(m_1+m_2), Du \right\rangle m_2 \right\rangle _{{\mathbb {L}}^2_w}&\le \frac{1}{2\varepsilon ^2}\gamma ^4 C_\kappa ^4 |Dm_1(s) + Dm_2(s)|_{{\mathbb {L}}^\infty }^2 |u(s)|_{{\mathbb {L}}^2_w}^2 \\&\quad + \frac{1}{2} \varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2. \end{aligned}$$

Also,

$$\begin{aligned} - \gamma ^2 \left\langle u, \kappa ^2 |Dm_1|^2 u \right\rangle _{{\mathbb {L}}^2_w}(s) \le 0, \quad \forall s \in [0,T]. \end{aligned}$$

For the remaining term in $U_2$, we use integration-by-parts as in (93):

$$\begin{aligned}&\frac{1}{2} (1-\gamma ^2) \left\langle u, \kappa ^2 D^2 u \right\rangle _{{\mathbb {L}}^2_w} \\&\quad = \frac{1}{2}(\gamma ^2-1) \left\langle Du, D(\kappa ^2 u \rho _w) \right\rangle _{{\mathbb {L}}^2} \\&\quad = \frac{1}{2}(\gamma ^2-1) \left[ \left\langle Du, \kappa ^2 Du \right\rangle _{{\mathbb {L}}^2_w}(s) + \left\langle Du, \kappa ^2 u \rho _w' \rho _w^{-1} + 2 \kappa \kappa ' u \right\rangle _{{\mathbb {L}}^2_w}(s) \right] \\&\quad \le -\frac{1}{2} \sum _j q_j^2 |f_j Du(s)|^2_{{\mathbb {L}}^2_w} + \frac{1}{2}\gamma ^2 C_\kappa ^2 |Du(s)|^2_{{\mathbb {L}}^2_w} + \frac{1}{4\varepsilon ^2} C_\kappa ^4 \left( 1-\gamma ^2 \right) ^2(w^2+4) \ |u(s)|^2_{{\mathbb {L}}^2_w} \\&\qquad + \frac{1}{2}\varepsilon ^2 |Du(s)|^2_{{\mathbb {L}}^2_w}. \end{aligned}$$

Thus,

$$\begin{aligned} U_2(s)&\le \psi _2(s) |u(s)|_{{\mathbb {L}}^2_w}^2 + \frac{5}{2}\varepsilon ^2 |Du(s)|_{{\mathbb {L}}^2_w}^2 + \frac{1}{2}\gamma ^2 C_\kappa ^2 |Du(s)|^2_{{\mathbb {L}}^2_w} -\frac{1}{2} \sum _j q_j^2 |f_j Du(s)|^2_{{\mathbb {L}}^2_w}, \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \psi _2(s)&= \frac{1}{4 \varepsilon ^2} C_\kappa ^4 (1-\gamma ^2)^2(w^2+5) \\&\quad + \frac{1}{2\varepsilon ^2} \left( \gamma ^2(w^2+6) + \gamma ^2 |Dm_2(s)|_{{\mathbb {L}}^\infty }^2 + \gamma ^4 |Dm_1(s) + Dm_2(s)|_{{\mathbb {L}}^\infty }^2 \right) , \end{aligned} \end{aligned}$$

(97)

and by (96), $\int _0^T \psi _2(t) \ \textrm{d}t < \infty$, ${\mathbb {P}}$-a.s.

$\underline{\text { An estimate on } \,\, U_3:}$

$$\begin{aligned} U_3(s)&= \frac{1}{2} \sum _j q_j^2 \left| f_j \left( G(m_1) - G(m_2) \right) \right| _{{\mathbb {L}}^2_w}^2(s) \\&= \frac{1}{2} \sum _j q_j^2 \left| f_j \left( -Du(s) + \gamma u(s) \times Dm_1(s) + \gamma m_2(s) \times Du(s) \right) \right| _{{\mathbb {L}}^2_w}^2 (s), \end{aligned}$$

where for every $j \ge 1$,

$$\begin{aligned}&f_j^2 \left| G(m_1) - G(m_2) \right| ^2(s,x) \\&\quad = f_j^2 \left| -Du + \gamma u \times Dm_1 + \gamma m_2 \times Du \right| ^2(s,x) \\&\quad = f_j^2 \left( |Du|^2 + \gamma ^2 |m_2 \times Du|^2 + \gamma ^2 |u \times Dm_1|^2 \right. \\&\qquad \left. + 2 \gamma \left\langle -Du + \gamma m_2 \times Du ,u \times Dm_1 \right\rangle \right) (s,x) \\&\quad \le (1+\gamma ^2)|f_j Du(s,x)|^2 + q_j^2\gamma ^2 \ |Dm_1(s)|^2_{{\mathbb {L}}^\infty } \ |f_j u(s,x)|^2 \\&\qquad + \frac{2}{\varepsilon ^2} \gamma ^2 \left( 1+ \gamma ^2 \right) |Dm_1(s)|^2_{{\mathbb {L}}^\infty } \ |f_j^2 u(s,x)|^2 + \varepsilon ^2 |Du(s,x)|^2. \end{aligned}$$

Hence,

$$\begin{aligned} U_3(s)&\le \frac{1}{2} (1+\gamma ^2) \sum _j q_j^2 |f_j Du(s)|^2_{{\mathbb {L}}^2_w} + \gamma ^2 C_\kappa ^2\\&\quad \left( \frac{1}{2} + \frac{1}{\varepsilon ^2}(1+\gamma ^2) C_\kappa ^2 \right) \ |Dm_1(s)|^2_{{\mathbb {L}}^\infty } |u(s)|^2_{{\mathbb {L}}^2_w} \\&\quad + \frac{1}{2} \varepsilon ^2 |Du(s)|^2_{{\mathbb {L}}^2_w} \\&\le \psi _3(s) |u(s)|^2_{{\mathbb {L}}^2_w} + \frac{1}{2} \sum _j q_j^2 |f_j Du(s)|^2_{{\mathbb {L}}^2_w} \\&\quad + \frac{1}{2}\gamma ^2 C_\kappa ^2 |Du(s)|^2_{{\mathbb {L}}^2_w} + \frac{1}{2} \varepsilon ^2 |Du(s)|^2_{{\mathbb {L}}^2_w}, \end{aligned}$$

where the second term on the right-hand side cancels with the corresponding term in $U_2(s)$ and $\psi _3(s) = \gamma ^2 C_\kappa ^2 \left( \frac{1}{2} + \frac{1}{\varepsilon ^2}(1+\gamma ^2) C_\kappa ^2 \right) |Dm_1(s)|^2_{{\mathbb {L}}^\infty }$ is similarly integrable ${\mathbb {P}}$-a.s.

We have

$$\begin{aligned} U_1(s) + U_2(s) + U_3(s)&\le \left( \psi _1(s)+ \psi _2(s) + \psi _3(s) \right) |u(s)|_{{\mathbb {L}}^2_w}^2 \\&\quad + \left( 6\varepsilon ^2 + \gamma ^2 C_\kappa ^2 - \alpha \right) |Du(s)|_{{\mathbb {L}}^2_w}^2. \end{aligned}$$

We can choose a sufficiently small $\varepsilon > 0$ such that under the assumption (21),

$$\begin{aligned} \left( 6 \varepsilon ^2 + \gamma ^2 C_\kappa ^2 - \alpha \right) < 0, \end{aligned}$$

which implies

$$\begin{aligned} U_1(s) + U_2(s) + U_3(s)&\le \left( \psi _1(s)+ \psi _2(s) + \psi _3(s) \right) |u(s)|_{{\mathbb {L}}^2_w}^2 = \psi (s) \ |u(s)|^2_{{\mathbb {L}}^2_s}. \end{aligned}$$

Therefore, by (89),

$$\begin{aligned} \frac{1}{2}d|u(t)|_{{\mathbb {L}}^2_w}^2 \le \psi (t) |u(t)|_{{\mathbb {L}}^2_w}^2 \ \textrm{d}t + \left\langle u(t), \left[ G(m_1(t)) - G(m_2(t)) \right] \textrm{d}W(t) \right\rangle _{{\mathbb {L}}^2_w}. \end{aligned}$$

Define the process Y by

$$\begin{aligned} Y(t) := \frac{1}{2}|u(t)|_{{\mathbb {L}}^2_w}^2 e^{-2\int _0^t \psi (s) \ \textrm{d}s}, \quad t \in [0,T]. \end{aligned}$$

Then,

$$\begin{aligned} dY(t)&= \left\langle \frac{1}{2}d|u(t)|_{{\mathbb {L}}^2_w}^2, e^{-2\int _0^t \psi (s) \ \textrm{d}s} \right\rangle + \left\langle \frac{1}{2}|u(t)|_{{\mathbb {L}}^2_w}^2, de^{-2\int _0^t \psi (s) \ \textrm{d}s} \right\rangle \\&\quad + \left\langle \frac{1}{2}d|u(t)|_{{\mathbb {L}}^2_w}^2, de^{-2\int _0^t \psi (s) \ \textrm{d}s} \right\rangle \\&\le e^{-2\int _0^t \psi (s) \ \textrm{d}s} \left\langle u(t), \left[ G(m_1(t)) - G(m_2(t)) \right] \textrm{d}W(t) \right\rangle _{{\mathbb {L}}^2_w}. \end{aligned}$$

Since $|u(t)|_{{\mathbb {L}}^\infty } \le 2$ ${\mathbb {P}}$-a.s. and there exists a constant C such that

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t \in [0,T]} \left( |Dm_1(t)|_{{\mathbb {L}}^2}^2 + |Dm_2(t)|_{{\mathbb {L}}^2}^2 \right) \right] \le C, \end{aligned}$$

the process

$$\begin{aligned} M(t) := \int _0^t e^{-2\int _0^s \psi (r) \ dr} \left\langle u(s), \left[ G(m_1(s)) - G(m_2(s)) \right] \textrm{d}W(s) \right\rangle _{{\mathbb {L}}^2_w}. \end{aligned}$$

is a martingale, and then

$$\begin{aligned} {\mathbb {E}}[Y(t)] \le Y(0) + {\mathbb {E}}[M(t)] = Y(0), \quad t \in [0,T]. \end{aligned}$$

By the definition of Y(t) , if $Y(0) = m_1(0) - m_2(0) = 0$, then

$$\begin{aligned} |u(t)|_{{\mathbb {L}}^2_w}^2 = 0, \quad {\mathbb {P}}\text {-a.s.} \end{aligned}$$

for $t \in [0,T]$, proving pathwise uniqueness of the solution of (8). By the Yamada–Watanabe Theorem, the uniqueness in law follows.

APPENDIX A

1.1 A.1 Some calculations in discrete spaces

• (a) discrete integration-by-parts:

  $$\begin{aligned} \sum _{x \in {\mathbb {Z}}_h} \langle u(x), \partial ^{h} w(x) \rangle = - \sum _{x \in {\mathbb {Z}}_h} \langle \partial ^{h} u^-(x), w(x) \rangle , \end{aligned}$$

  for $u,w \in {\mathbb {H}}_h^1 = \{ v \in {\mathbb {L}}_h^2: |\partial ^h v|_{{\mathbb {L}}_h^2} < \infty \}$ with appropriate decay properties. In particular,

  $$\begin{aligned} \left\langle \partial ^h u, \partial ^h w \right\rangle _{{\mathbb {L}}_h^2} = - \left\langle \Delta ^h u, w \right\rangle _{{\mathbb {L}}_h^2}. \end{aligned}$$

• (b) discrete expansion of $\langle u, \Delta ^h u \rangle$ and $\langle u, \partial ^h u \rangle$: for any u satisfying that $|u(x)|=1$ for all $x\in {\mathbb {Z}}_h$,

  $$\begin{aligned} \begin{aligned} \langle u(x), \Delta ^h u (x) \rangle&= - \frac{1}{2} \left( |\partial ^{h} u(x)|^2 + |\partial ^{h} u^-(x)|^2 \right) \le 0, \\ \langle u(x), \partial ^h u(x) \rangle&= -\frac{h}{2} |\partial ^h u(x)|^2 \le 0. \end{aligned} \end{aligned}$$

  (98)

• (c) product rule:

  $$\begin{aligned} \partial ^{h} (fu)&= (\partial ^{h} f) u(x) + f^+ \partial ^{h} u = (\partial ^{h} f) u^+ + f \partial ^{h} u(x) \end{aligned}$$

  for f scalar-valued and u vector-valued; similarly for f and u both scalar-valued, and for $\langle f, u \rangle$ and $u \times u$ when f, u are vector-valued.

• (d) $L_h^2$-norm of $\Delta ^h u$:

  $$\begin{aligned} |\Delta ^h u|_{{\mathbb {L}}_h^2} = |\partial ^{h} (\partial ^{h} u)^-|_{{\mathbb {L}}_h^2} = |\partial ^{h}(\partial ^{h} u)|_{{\mathbb {L}}_h^2}. \end{aligned}$$

Lemma A.1

[43, Chapter 1, Theorem 3] For $u^h: {\mathbb {Z}}_h \rightarrow {\mathbb {R}}$,

$$\begin{aligned} \left| (\partial ^{h})^k u^h \right| _{{\mathbb {L}}_h^p} \le C |u^h|_{{\mathbb {L}}_h^2}^{1- \frac{k+ \frac{1}{2} - \frac{1}{p}}{n}} \left| (\partial ^{h})^n u^h \right| _{{\mathbb {L}}_h^2}^{\frac{k+\frac{1}{2} - \frac{1}{p}}{n}}, \end{aligned}$$

for $p \in [2,\infty ]$, $k \in [0,n)$ and C is a constant independent of $u^h$.

1.2 A.2 Some tightness results

Lemma A.2

[17, Theorem 2.1] Let $B_0 \subset B \subset B_1$ be Banach spaces, $B_0$ and $B_1$ reflexive, with compact embedding of $B_0$ in B. Let $p \in (1,\infty )$ and $\alpha \in (0,1)$ be given. Let X be the space

$$\begin{aligned} X = L^p(0,T; B_0) \cap W^{\alpha ,p}(0,T;B_1) \end{aligned}$$

endowed with the natural norm. Then the embedding of X in $L^p(0,T; B)$ is compact.

Lemma A.3

[17, Theorem 2.2] If $B_1 \subset {\widetilde{B}}$ are two Banach spaces with compact embedding, and the real numbers $\alpha \in (0,1)$, $p>1$ satisfy $\alpha p > 1$, then the space $W^{\alpha ,p}(0,T; B_1)$ is compactly embedded into ${\mathcal {C}}([0,T]; {\widetilde{B}})$.

Lemma A.4

[37, Corollary 19] Let I be an either bounded or unbounded interval of ${\mathbb {R}}$. Let E be a Banach space. Suppose $s \ge r$, $p \le q$ and $s- \frac{1}{p} \ge r - \frac{1}{q}$ for $0< r \le s <1$ and $1 \le p \le q \le \infty$. Then,

$$\begin{aligned} W^{s,p}(I; E) \hookrightarrow W^{r,q}(I; E). \end{aligned}$$

In addition, we verify the continuous embedding

$$\begin{aligned} W^{\alpha , 4}(0,T; {\mathbb {L}}^2_w) \hookrightarrow W^{\beta ,2}(0,T; {\mathbb {L}}^2_w), \quad w \ge 1, \ \alpha \in \left( \frac{1}{4}, \frac{1}{2} \right) ,\ \beta = \alpha - \frac{1}{4}. \end{aligned}$$

(99)

Indeed, for $u \in W^{\alpha ,4}(0,T; {\mathbb {L}}^2_w)$,

$$\begin{aligned} |u|_{W^{\beta ,2}(0,T;{\mathbb {L}}^2_w)}&= \left[ \int _0^T |u(t)|_{{\mathbb {L}}^2_w}^2 \ \textrm{d}t + \int _0^T \int _0^T \frac{|u(t)-u(s)|_{{\mathbb {L}}^2_w}^2}{|t-s|^{1+2\beta }} \ \textrm{d}t \ \textrm{d}s \right] ^\frac{1}{2} \\&\le \left[ \left( \int _0^T |u(t)|_{{\mathbb {L}}^2_w}^4 \ \textrm{d}t\right) ^{\frac{1}{2}} T^\frac{1}{2} + \left( \int _0^T \int _0^T \frac{|u(t)-u(s)|_{{\mathbb {L}}^2_w}^4}{|t-s|^{2+4\beta }} \ \textrm{d}t \ \textrm{d}s \right) ^{\frac{1}{2}} T \right] ^\frac{1}{2} \\&\le \left[ 2 T \int _0^T |u(t)|_{{\mathbb {L}}^2_w}^4 \ \textrm{d}t + 2 T^2 \int _0^T \int _0^T \frac{|u(t)-u(s)|_{{\mathbb {L}}^2_w}^4}{|t-s|^{2+4\beta }} \ \textrm{d}t \ \textrm{d}s \right] ^{\frac{1}{4}} \\&\le \left( 2 \max \{ T, T^2 \} \right) ^\frac{1}{4} \left[ \int _0^T |u(t)|_{{\mathbb {L}}^2_w}^4 \ \textrm{d}t + \int _0^T \int _0^T \frac{|u(t)-u(s)|_{{\mathbb {L}}^2_w}^4}{|t-s|^{1+4\alpha }} \ \textrm{d}t \ \textrm{d}s \right] ^\frac{1}{4} \\&= \left( 2 \max \{ T, T^2 \} \right) ^\frac{1}{4} |u|_{W^{\alpha ,4}(0,T; {\mathbb {L}}^2_w)}, \end{aligned}$$

where the second inequality holds by $\sqrt{a} + \sqrt{b} \le \sqrt{2a + 2b}$ for $a,b \ge 0$.