On the energy transfer to high frequencies in the damped/driven nonlinear Schrödinger equation

Abstract

We consider a damped/driven nonlinear Schrödinger equation in $\mathbb {R}^n$, where n is arbitrary, ${\mathbb {E}}u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad \nu >0,$ under odd periodic boundary conditions. Here $\eta (t,x)$ is a random force which is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $\Vert u(t)\Vert _m^2 \le C\nu ^{-m},$ uniformly in $t\ge 0$ and $\nu >0$. In this work we prove that for small $\nu >0$ and any initial data, with large probability the Sobolev norms $\Vert u(t,\cdot )\Vert _m$ with $m>2$ become large at least to the order of $\nu ^{-\kappa _{n,m}}$ with $\kappa _{n,m}>0$, on time intervals of order $\mathcal {O}(\frac{1}{\nu })$. It proves that solutions of the equation develop short space-scale of order $\nu$ to a positive degree, and rigorously establishes the (direct) cascade of energy for the equation.

1 Introduction

In this work we study a damped/driven nonlinear Schrödinger equation

$$\begin{aligned} u_t-\nu \Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x), \quad x\in {\mathbb {R}}^{n}, \end{aligned}$$

(1.1)

i.e. a CGL equation without linear dispersion, with cubic Hamiltonian nonlinearity and a random forcing. The dimension n is any, $0<\nu \le 1$ is the viscosity constant and the random force $\eta$ is white in time t and regular in x. The equation is considered under the odd periodic boundary conditions,

$$\begin{aligned} u(t,\dots ,x_j,\dots )=u(t,\dots ,x_j+2\pi ,\dots )=-u(t,\dots ,x_j+\pi ,\dots ),\quad j=1,\dots ,n. \end{aligned}$$

The latter implies that u vanishes on the boundary of the cube of half-periods $K^{n}= [0, \pi ]^n$,

$$\begin{aligned} u\mid _{\partial K^{n}} =0. \end{aligned}$$

We denote by $\{\varphi _d(\cdot ),\; d=(d_1,\dots ,d_n)\in {\mathbb {N}}^n\}$ the trigonometric basis in the space of odd periodic functions,

$$\begin{aligned} \varphi _d(x)=(\tfrac{2}{\pi })^{\frac{n}{2}}\sin (d_1x_1)\cdots \sin (d_nx_n). \end{aligned}$$

The basis is orthonormal with respect to the normalised scalar product $\langle \!\langle \cdot ,\cdot \rangle \!\rangle$ in $L_2(K^{n}, \pi ^{-n} dx)$,

$$\begin{aligned} \langle \!\langle u,v\rangle \!\rangle = \int _{K^n} \langle u(x), v(x)\rangle \pi ^{-n} dx, \end{aligned}$$

(1.2)

where $\langle \cdot , \cdot \rangle$ is the real scalar product in ${\mathbb {C}}$, $\langle u, v \rangle =\mathfrak {R}u{\bar{v}}$. It is formed by eigenfunctions of the Laplacian:

$$\begin{aligned} (-\Delta )\varphi _d=|d|^2\varphi _d. \end{aligned}$$

The force $\eta (t,x)$ is a random field of the form

$$\begin{aligned} \eta (t,x)=\frac{\partial }{\partial t}\xi (t,x),\quad \xi (t,x)=\sum _{d\in {\mathbb {N}}^n}b_d\beta _d(t)\varphi _d(x). \end{aligned}$$

(1.3)

Here $\beta _d(t)=\beta ^R_d(t)+i\beta ^I_d(t)$, where $\beta _d^R(t)$, $\beta _d^I(t)$ are independent real-valued standard Brownian motions, defined on a complete probability space $(\Omega ,{\mathcal {F}},{\mathbb {P}})$ with a filtration $\{{\mathcal {F}}_t;t\geqslant 0\}$. The set of real numbers $\{b_d,\;d\in {\mathbb {N}}^n\}$ is assumed to form a non-zero sequence, satisfying

$$\begin{aligned} 0<B_{m_*}<\infty , \quad m_*= \min \{ m\in {\mathbb {Z}}: m>n/2\}, \end{aligned}$$

(1.4)

where for a real number k we set

$$\begin{aligned} B_k:=\sum _{d\in {\mathbb {N}}^n}|d|^{2k} |b_d|^2\le \infty . \end{aligned}$$

For $m\ge 0$ we denote by $H^m$ the Sobolev space of order m, formed by complex odd periodic functions, equipped with the homogeneous norm,

$$\begin{aligned}\Vert u\Vert _m=\Vert (-\Delta )^{\frac{m}{2}}u\Vert _0,\end{aligned}$$

where $\Vert \cdot \Vert _0$ is the $L^2$-norm on $K^{n}$, $\Vert u\Vert ^2_0 = \langle \!\langle u,u\rangle \!\rangle$ (see (1.2)). If we write $u\in H^m$ as Fourier series, $u(x)=\sum _{d\in {\mathbb {N}}^n}u_d\varphi _d(x),$ then $\Vert u\Vert _m^2=\sum _{d\in {\mathbb {N}}^n}|d|^{2m}| u_d|^2.$

Equation (1.1) with small $\nu$ belongs to a group of equations, describing turbulence in the CGL equations. These equations have got quite a lot of attention in physical literature as models for turbulence in various media, e.g. see [3, Chapter 5]. In particular – as a natural model for hydrodynamical turbulence since Eq. (1.1) is obtained from the Navier-Stokes system by replacing the Euler term $(u\cdot \nabla ) u$, which is a quadratic Hamiltonian nonlinearity, by $i|u|^2u$, which is a cubic Hamiltonian nonlinearity, see [13].

The global solvability of Eq. (1.1) for any space dimension n is established in [8, 10]. It is proved there that if

$$\begin{aligned} u(0,x)=u_0(x), \end{aligned}$$

(1.5)

where $u_0\in H^m\cap C(K^n)$, $m\in {\mathbb {N}}$, and if $B_m<\infty$, then the problem (1.1), (1.5) has a unique strong solution u(t, x) in $H^m$ which we write as $u(t,x;u_0)$, or $u(t;u_0)$, or $u_\nu (t;u_0)$. Its norm satisfies

$$\begin{aligned} {\mathbb {E}}\Vert u(t;u_0)\Vert _m^2 \le C_m \nu ^{-m},\quad t\ge 0, \end{aligned}$$

where $C_m$ depends on $\Vert u_0\Vert _m, |u_0|_\infty$ and $B_m, B_{m_*}$. Furthermore, denoting by $C_0(K^{n})$ the space of continuous complex functions on $K^{n}$, vanishing at $\partial K^{n}$, we have that the solutions u(t, x) define a Markov process in $C_0(K^{n})$. Moreover, if the noise $\eta (t,\cdot )$ is non-degenerate in the sense that in (1.3) all coefficients $b_d$ are non-zero, then this process is mixing.¹

Our goal is to study the growth of higher Sobolev norms for solutions of Eq. (1.1) as $\nu \rightarrow 0$ on time intervals of order ${\mathcal {O}}(\frac{1}{\nu })$. The main result of this work is the following.

Theorem 1

For any real number $m>2$, in addition to (1.4), assume that $B_m<\infty$. Then there exists $\kappa _{n,m}>0$ such that for every fixed quadruple $(\delta ,\kappa ,{\mathscr {K}}, T_0)$, where

$$\begin{aligned} \kappa \in (0,\kappa _{n,m}),\quad \delta \in (0,\tfrac{1}{8}), \quad {\mathscr {K}}, T_0>0, \end{aligned}$$

there exists a $\nu _0>0$ with the property that if $0<\nu \le \nu _0$, then for every $u_0\in H^m \cap C_0(K^{n})$, satisfying

$$\begin{aligned} |u_0|_\infty \leqslant {\mathscr {K}}, \quad \Vert u_0\Vert _m \le \nu ^{-\kappa m}, \end{aligned}$$

(1.6)

the solution $u(t,x;u_0)$ is such that

• (1)
  $$\begin{aligned} {\mathbb {P}}\big \{ \sup _{t \in [t_0, t_0+ T_0\nu ^{-1}]} \Vert u_\nu ^\omega (t)\Vert _m> \nu ^{-m\kappa }\big \} \ge 1-\delta , \quad \forall \, t_0\ge 0. \end{aligned}$$

• (2) If m is an integer, $m\ge 3$, then a possible choice of $\kappa _{n,m}$ is $\kappa _{n,m}= \tfrac{1}{35}$, and there exists $C\ge 1$, depending on $\kappa < \tfrac{1}{35}$, ${\mathscr {K}}, m,B_{m_*}$ and $B_m$, such that

  $$\begin{aligned} C^{-1} \nu ^{-2m\kappa +1}\leqslant {\mathbb {E}}\left( \nu \int _{t_0}^{t_0+\nu ^{-1}}\Vert u_\nu (s)\Vert _m^2ds\right) \leqslant C \nu ^{-m}, \quad \forall \, t_0\ge 0. \end{aligned}$$

  (1.7)

A similar result holds for the classical $C^k$-norms of solutions:

Proposition 2

For any integer $m \ge 2$ in addition to (1.4) assume that $B_m<\infty$. Then for every fixed triplet $K, {\mathcal {K}}, T_0>0$ and any $0<\kappa <1/16$ we have

$$\begin{aligned} {\mathbb {P}}\left\{ \sup _{t \in [t_0, t_0+ T_0\nu ^{-1}]} |u_\nu ^\omega (t;u_0)|_{C^m} > K \nu ^{-m\kappa }\right\} \rightarrow 1 \quad \text {as} \quad \nu \rightarrow 0, \end{aligned}$$

(1.8)

for each $t_0\ge 0$, if $u_0$ satisfies $|u_0|_\infty \le {\mathcal {K}}$, $|u_0|_{C^m} \le \nu ^{-\kappa m}$. The rate of convergence depends only on the triplet and $\kappa$.

For a proof of this result see the extended version of our work [6]. Due to (1.8), for any $m>2+n/2$ we have

$$\begin{aligned} {\mathbb {P}}\left\{ \sup _{T_0 \le t \le t_0+{T_0}{\nu ^{-1}}} \Vert u(t)\Vert _{m} \ge K \nu ^{- \lfloor m-\frac{n}{2} \rfloor \kappa } \right\} \rightarrow 1\quad \text {as} \quad \nu \rightarrow 0, \end{aligned}$$

for every $K>0$ and $0<\kappa <1/16$, where for $a\in {\mathbb {R}}$ we denote $\lfloor a \rfloor = \max \{ n\in {\mathbb {Z}}: n<a\}.$ This improves the first assertion of Theorem 1 for large m.

We have the following two corollaries from Theorem 1, valid if the Markov process defined by the Eq. (1.1) is mixing:

Corollary 3

Assume that $B_m<\infty$ for all m and $b_d\ne 0$ for all d. Then Eq. (1.1) is mixing and for any $\kappa <1/35$ and $0<\nu \le \nu _0$ its unique stationary measure $\mu _\nu$ satisfies

$$\begin{aligned} C^{-1} \nu ^{-2m\kappa +1} \le \int \Vert u\Vert _m^2\mu _\nu (du) \le C\nu ^{-m}, \quad 3\le m\in {\mathbb {N}}. \end{aligned}$$

(1.9)

Here C and $\nu _0$ are as in Theorem 1.

Corollary 4

Under the assumptions of Corollary 3, for any $u_0\in C^\infty$ we have

$$\begin{aligned} \tfrac{1}{2} C^{-1} \nu ^{-2m\kappa +1} \le {\mathbb {E}}\Vert u (s;u_0)\Vert _m^2 \le 2C\nu ^{-m}, \quad 3\le m\in {\mathbb {N}}, \end{aligned}$$

if $s\ge T(\nu , u_0, \kappa , B_m, B_{m_*})$, where C is the same as in (1.9).

Theorem 1 rigorously establishes the energy cascade to high frequencies for solutions of Eq. (1.1) with small $\nu$. Indeed, if $u_0(x)$ and $\eta (t,x)$ are smooth functions of x (or even trigonometric polynomials of x), then in view of (1.7) for $0<\nu \ll 1$ and $t > rsim \nu ^{-1}$ a substantial part of the energy $\frac{1}{2} \sum |u_d(t)|^2$ of a solution $u(t,x;u_0)$ is carried by high modes $u_d$ with $|d|\gg 1$. Relation (1.7) (valid for all integer $m\ge 3$) also means that the averaged in time space-scale $l_x$ of solutions for (1.1) satisfies $l_x \in [\nu ^{1/2}, \nu ^{1/35}],$ and goes to zero with $\nu$ (see [1, 9]). We recall that the energy cascade to high frequencies and formation of short space-scale is the driving force of the Kolmogorov theory of turbulence, see [5].

We mention that in the work [12] the stochastic CGL equation

$$\begin{aligned} u_t-(\nu +i)\Delta u+i|u|^2u=\sqrt{\nu }\eta (t,x),\quad 0<\nu \le 1, \end{aligned}$$

(1.10)

with linear dispersion and white in time random force $\eta$ as in (1.3) was considered under the odd periodic boundary conditions, and the inviscid limiting dynamics as $\nu \rightarrow 0$ was examined. However, since the limiting Eq. (1.10)${}_{\nu =0}$ is a regular PDE in difference with the Eq. (1.1)${}_{\nu =0}$, the results on the inviscid limit in [12] differ in spirit from those in our work, and we do not discuss them now.

Deterministic versions of the result of Theorem 1 for Eq. (1.1) with $\eta =0$, where $\nu$ is a small non-zero complex number such that $\mathfrak {R}\nu \ge 0$ and $\mathfrak {I}\nu \le 0$ are known, see [9]. In particular, if $\nu$ is a positive real number and $u_0$ is a smooth function of order one, then for any integer $m\ge 4$ a solution $u_\nu (t,x;u_0)$ satisfies estimates (1.7) with the averaging $\nu {\mathbb {E}}\int _t^{t+\nu ^{-1}}\! \!\dots ds$ replaced by $\nu ^{1/3} \int _0^{\nu ^{-1/3}}\! \!\dots ds,$ with the same upper bound and with the lower bound $C_m \nu ^{-\kappa _m m}$, where $\kappa _m\rightarrow 1/3$ as $m\rightarrow \infty$. Moreover, it was then shown in [2] that the lower bounds remain true with $\kappa =1/3$, and that the estimates $\sup _{t\in [0, |\nu |^{-1/3}]} \Vert u(t)\Vert _{C^m} \ge C_m |\nu |^{-m/3}, m\ge 2,$ hold for smooth solutions of Eq. (1.1) with $\eta =0$ and any non-zero complex “viscosity” $\nu$.

The better quality of the lower bounds for solutions of the deterministic equations is due to an extra difficulty which occurs in the stochastic case: when time grows, simultaneously with increasing of high Sobolev norms of a solution, its $L_2$-norm may decrease, which accordingly would weaken the mechanism, responding for the energy transfer to high modes. Significant part of the proof of Theorem 1 is devoted to demonstration that the $L_2$-norm of a solution cannot go down without sending up the second Sobolev norm.

If $\eta =0$ and $\nu =i\delta \in i{\mathbb {R}}$, then (1.1) is a Hamiltonian PDE (the defocusing Schrödinger equation), and the $L_2$-norm is its integral of motion. If this integral is of order one, then the results of [9] (see there Appendix 3) imply that at some point of each time-interval of order $\delta ^{-1/3}$ the $C^m$-norm of a corresponding solution will become $\ > rsim \delta ^{-m\kappa }$ if $m\ge 2$, for any $\kappa <1/3$. Furthermore, if $n=2$ and $\delta =1$, then due to [4] for $m>1$ and any $M>1$ there exists a $T=T(m,M)$ and a smooth $u_0(x)$ such that $\Vert u_0\Vert _m< M^{-1}$ and $\Vert u(T;u_0)\Vert _m>M$.

The paper is organized as follows. In Sect. 2, we recall the results from [8, 10] on solutions of the Eq. (1.1). Next we show in Sect. 3 that if the noise $\eta$ is non-degenerate, the $L^2$-norm of a solution of Eq. (1.1) cannot stay too small on time intervals of order ${\mathcal {O}}(\frac{1}{\nu })$ with high probability, unless its $H^2$-norm gets very large

(see Lemma 12). Then in Sect. 4 we derive from this fact the assertion (1) of Theorem 1. We prove assertion (2) and both corollaries in Sect. 5.

Constants in estimates never depend on $\nu$, unless otherwise stated. For a metric space M we denote by ${\mathcal {B}}(M)$ the Borel $\sigma$-algebra on M, and by ${{\mathcal {P}}}(M)$ – the space of probability Borel measures on M. By ${\mathcal {D}}(\xi )$ we denote the law of a r.v. $\xi$, and by $|\cdot |_p$ – the norm in $L_p(K^n)$.

2 Solutions and estimates

Strong solutions for the Eq. (1.1) are defined in the usual way:

Definition 5

Let $(\Omega , {\mathcal {F}},\{{\mathcal {F}}_t\}_{t\geqslant 0}, {\mathbb {P}})$ be the filtered probability space as in the introduction. Let $u_0$ in (1.5) be a r.v., measurable in ${\mathcal {F}}_0$ and independent from the Wiener process $\xi$ (e.g., $u_0(x)$ may be a non-random function). Then a random process $u(t)=u(t;u_0)\in C_0(K^{n})$, $t\in [0,T]$, adapted to the filtration, is called a strong solution of (1.1), (1.5), if

• (1) a.s. its trajectories u(t) belong to the space

  $$\begin{aligned} {\mathcal {H}}([0,T]):=C([0,T],C_0(K^{n}))\cap L^2([0,T], H^1); \end{aligned}$$

• (2) we have

  $$\begin{aligned} u(t)=u_0+\int _0^t(\nu \Delta u-i|u|^2u)ds+\sqrt{\nu }\,\xi (t),\quad \forall t\in [0,T], \quad a.s., \end{aligned}$$

  where both sides are regarded as elements of $H^{-1}$.

If (1)-(2) hold for every $T<\infty$, then u(t) is a strong solution for $t\in [0,\infty )$. In this case a.s. $u\in C([0,\infty ),C_0(K^{n}))\cap L_{loc}^2([0,\infty ), H^1).$

Everywhere below when we talk about solutions for the problem (1.1), (1.5) we assume that the r.v. $u_0$ is as in the definition above.

The global well-posedness of Eq. (1.1) was established in [8, 10]:

Theorem 6

For any $u_0\in C_0(K^{n})$ the problem (1.1), (1.5) has a unique strong solution $u^\omega (t,x;u_0)$, $t\ge 0$. The family of solutions $\{ u^\omega (t;u_0)\}$ defines in the space $C_0(K^{n})$ a Fellerian Markov process.

In [8, 10] the theorem above is proved when (1.4) is replaced by the weaker assumption $B_*<\infty$, where $\ B_* =\sum |b_d|$ (note that $B_*\le C_n B_{m_*}^{1/2}$).

The transition probability for the obtained Markov process in $C_0(K^{n})$ is

$$\begin{aligned} P_t(u,\Gamma )={\mathbb {P}}\{u(t;u)\in \Gamma \}, \quad u\in C_0(K^{n}),\; \Gamma \in {\mathscr {B}}(C_0(K^{n})), \end{aligned}$$

and the corresponding Markov semigroup in the space ${\mathscr {P}}(C_0(K^{n}))$ of Borel measures on $C_0(K^{n})$ is formed by the operators $\{ {\mathcal {B}}_t^*, t\ge 0\}$,

$$\begin{aligned} {\mathcal {B}}_t^*\mu (\Gamma )=\int _{C_0(K^{n})}P_t(u,\Gamma )\mu (du),\quad t\in {\mathbb {R}}. \end{aligned}$$

Then ${\mathcal {B}}_t^*\mu = {\mathcal {D}}u(t;u_0)$ if $u_0$ is a r.v., independent from $\xi$ and such that ${\mathcal {D}}(u_0)=\mu$.

Introducing the slow time $\tau =\nu t$ and denoting $v(\tau ,x)=u(\frac{\tau }{\nu },x)$, we rewrite Eq. (1.1) in the following form, more convenient for some calculations:

$$\begin{aligned} \frac{\partial v}{\partial \tau }-\Delta v+i\nu ^{-1}|v|^2v={\tilde{\eta }}(\tau ,x), \end{aligned}$$

(2.1)

where

$$\begin{aligned}{\tilde{\eta }}(\tau ,x)=\frac{\partial }{\partial \tau }{\tilde{\xi }}(\tau ,x),\quad {\tilde{\xi }}(\tau ,x)=\sum _{d\in {\mathbb {N}}^n}b_d{\tilde{\beta }}_d(\tau )\varphi _d(x),\end{aligned}$$

and ${\tilde{\beta }}_d(\tau ):=\nu ^{1/2}\beta _d(\tau \nu ^{-1})$, $d\in {\mathbb {N}}^d$, is another set of independent standard complex Brownian motions.

Let $\Upsilon \in C^\infty ({\mathbb {R}})$ be any smooth function such

$$\begin{aligned}\Upsilon (r)={\left\{ \begin{array}{ll}0 ,&{}\text { for }r\leqslant \frac{1}{4};\\ r, &{}\text { for }r\geqslant \frac{1}{2}. \end{array}\right. }\end{aligned}$$

Writing $v\in {\mathbb {C}}$ in the polar form $v=re^{i\Phi }$, where $r=|v|$, and recalling that $\langle \cdot ,\cdot \rangle$ stands for the real scalar product in ${\mathbb {C}}$, we apply Itô’s formula to $\Upsilon (|v|)$ and obtain that the process $\Upsilon (\tau ):=\Upsilon (|v(\tau )|)$ satisfies

$$\begin{aligned} \begin{aligned} \Upsilon (\tau )&=\Upsilon _0+\int _0^\tau \Big [\Upsilon '(r)(\nabla r-r|\nabla \Phi |^2)\\&\quad +\frac{1}{2}\sum _{d\in {\mathbb {N}}^n}b_d^2\Big (\Upsilon ''(r)\langle e^{i\Phi },\varphi _d\rangle ^2+ \Upsilon '(r)\frac{1}{r}(|\varphi _d|^2-\langle e^{i\Phi },\varphi _d\rangle ^2)\Big )\Big ]ds+{\mathbb {W}}(\tau ),\end{aligned} \end{aligned}$$

(2.2)

where $\Upsilon _0=\Upsilon (|v(0)|)$ and ${\mathbb {W}}(\tau )$ is the stochastic integral

$$\begin{aligned}{\mathbb {W}}(\tau )=\sum _{d\in {\mathbb {N}}^n}\int _0^\tau \Upsilon '(r)b_d\varphi _d\langle e^{i\Phi }, d{\tilde{\beta }}_d(s)\rangle .\end{aligned}$$

In [10] Eq. (2.1) is considered with $\nu =1$ and, following [8], the norm $|v(t)|_\infty$ of a solution v is estimated via $\Upsilon (t)$ (since $|v| \le \Upsilon +1/2$). But the nonlinear term $i\nu ^{-1}|v|^2v$ does not contribute to Eq. (2.2), which is the same as the $\Upsilon$-equation (2.3) in [10] (and as the corresponding equation in [8, Section 3.1]). So the estimates on $|\Upsilon (t)|_\infty$ and the resulting estimates on $|v(t)|_\infty$, obtained in [10], remain true for solutions of (2.1) with any $\nu$. Thus we get the following upper bound for quadratic exponential moments of the $L_\infty$-norms of solutions:²

Theorem 7

For any $T>0$ there are constants $c_*>0$ and $C>0$, depending only on $B_*$ and T, such that for any r.v. $v_0^\omega \in C_0(K^{n})$ as in Definition 5, any $\tau \geqslant 0$ and any $c\in (0,c_*]$, a solution $v(\tau ; v_0)$ of Eq. (2.1) satisfies

$$\begin{aligned} {\mathbb {E}}\exp (c \sup _{\tau \leqslant s\leqslant \tau +T}|v(s)|_\infty ^2)\leqslant C\, {\mathbb {E}}\exp (5c\ |v_0|^2_\infty )\le \infty . \end{aligned}$$

(2.3)

In [10] the result above is proved for a deterministic initial data $v_0$. The theorem’s assertion follows by averaging the result of [10] in $v_0^\omega$.

The estimate (2.3) is crucial for derivation of further properties of solutions, including the given below upper bounds for their Sobolev norms, obtained in the work [8]. Since the scaling of the equation in [8] differs from that in (2.1) and the result there is a bit less general than in the theorem below, a sketch of the proof is given in Appendix B.

Theorem 8

Assume that $B_m<\infty$ for some $m\in {\mathbb {N}}$, and $v_0=v_0^\nu \in H^m\cap C_0(K^n)$ satisfies

$$\begin{aligned} |v_0|_\infty \le M, \quad \Vert v_0\Vert _m \le M_m \nu ^{-m}, \quad 0<\nu \le 1. \end{aligned}$$

Then

$$\begin{aligned} {\mathbb {E}}\Vert v(\tau ;v_0)\Vert _m^2\leqslant C_{m}\nu ^{-m},\quad \forall \tau \in [0,\infty ), \end{aligned}$$

(2.4)

where $C_{M,m}$ also depends on M, $M_m$ and $B_m$, $B_{m_*}$.

Neglecting the dependence on $\nu$, we have that if $B_m<\infty$, $m\in {\mathbb {N}}$, and a r.v. $v_0^\omega \in H^m\cap C_0(K^n)$ satisfies ${\mathbb {E}}\Vert v_0\Vert _m^2<\infty$ and ${\mathbb {E}}\exp (c\ |v_0|^2_\infty )<\infty$ for some $c>0$, then Eq. (2.1) has a solution, equal $v_0$ at $t=0$, such that

$$\begin{aligned}&{\mathbb {E}}\Vert v(\tau ;v_0)\Vert _m^2 \le e^{- t} {\mathbb {E}}\Vert v_0\Vert _m^2 +C, \quad \tau \ge 0, \end{aligned}$$

(2.5)

$$\begin{aligned}&{\mathbb {E}}\sup _{0\le \tau \le T} \Vert v(\tau ;v_0)\Vert _m^2 \le C' , \end{aligned}$$

(2.6)

where $C>0$ depend on $c, \nu , {\mathbb {E}}\exp (c\ |v_0|^2_\infty ), B_{m_*}$ and $B_m$, while $C'$ also depends on ${\mathbb {E}}\Vert v_0\Vert _m^2<\infty$ and T. See Appendix B.

As it is shown in [10], the estimate (2.3) jointly with an abstract theorem from [11], imply that under a mild nondegeneracy assumption on the random force the Markov process in the space $C_0(K^{n})$, constructed in Theorem 6, is mixing:

Theorem 9

For each $\nu >0$, there is an integer $N=N(B_*,\nu )> 0$ such that if $b_d\ne 0$ for $|d|\leqslant N$, then the Eq. (1.1) is mixing. I.e. it has a unique stationary measure $\mu _\nu \in {\mathscr {P}}(C_0(K^{n}))$, and for any probability measure $\lambda \in {\mathscr {P}}(C_0(K^{n}))$ we have ${\mathcal {B}}^*_t\lambda \rightharpoonup \mu _\nu$ as $t\rightarrow \infty$.

Under the assumption of Theorem 8, for any $u_0\in H^m$ the law ${\mathcal {D}}u(t;u_0)$ of a solution $u(t;u_0)$ is a measure in $H^m$. The mixing property in Theorem 9 and (2.4) easily imply

Corollary 10

If under the assumptions of Theorem 9$B_m<\infty$ for some $m\in {\mathbb {N}}$ and $u_0\in H^m$, then ${\mathcal {D}}(u(t;u_0)) \rightharpoonup \mu _\nu$ in ${\mathcal {P}}(H^m)$.

In view of Theorems  7, 8 with $v_0=0$ and the established mixing, we have:

Corollary 11

Under the assumptions of Theorem 9, if $v^{st}(\tau )$ is the stationary solution of the equation, then

$$\begin{aligned} {\mathbb {E}}\exp (c_*\sup _{\tau \leqslant s\leqslant \tau +T}|v^{st}(s)|_\infty ^2)\leqslant {\mathcal {C}}, \end{aligned}$$

where the constant ${\mathcal {C}}>0$ depends only on T and $B_*$. If in addition $B_m<\infty$ for some $m\in {\mathbb {N}}\cup \{0\}$, then ${\mathbb {E}}\Vert v^{st}(\tau )\Vert _m^2\leqslant C_{m}\nu ^{-m},$ where $C_m$ depends on $B_*$ and $B_m$.

Finally we note that applying Itô’s formula to $\Vert v^{st}(\tau )\Vert _0^2$, where $v^{st}$ is a stationary solution of (2.1), and taking the expectation we get the balance relation

$$\begin{aligned} {\mathbb {E}}\Vert v^{st}(\tau )\Vert ^2_1 = B_0. \end{aligned}$$

(2.7)

We cannot prove that ${\mathbb {E}}\Vert v^{st}(\tau )\Vert _0^2 \ge B'>0$ for some $\nu$-independent constant $B'$, and cannot bound from below the energy $\tfrac{1}{2} {\mathbb {E}}\Vert v(\tau ; v_0)\Vert ^2_0$ of a solution v by a positive $\nu$-independent quantity. Instead in next section we get a weaker conditional lower bound on the energies of solutions.

3 Conditional lower bound for the $L^2$-norm of solutions

In this section we prove the following result:

Lemma 12

Let $B_2<\infty$ and $u(\tau ;u_0)$, where $u_0\in H^2\cap C_0(K^n)$ is non-random, be a solution of Eq. (2.1). Take any constants $\chi >0, \Gamma \ge 1, \tau _0\ge 0,$ and define the stopping time

$$\begin{aligned} \tau _\Gamma :=\inf \{\tau \ge \tau _0 : \Vert u(\tau )\Vert _2\geqslant \Gamma \} \end{aligned}$$

(as usual, $\tau _\Gamma =\infty$ if the set under the $\inf$-sign is empty). Then

$$\begin{aligned} {\mathbb {E}} \int _{\tau _0}^{\tau \wedge \tau _\Gamma } {\mathbb {I}}_{[0,\chi ]} (\Vert u(s)\Vert _0)ds\leqslant 2(1+\tau -\tau _0)B_0^{-1} \chi \Gamma , \end{aligned}$$

(3.1)

for any $\tau >\tau _0$.

Proof

We establish the result by adapting the proof from [16] (also see [11, Theorem 5.2.12]) to non-stationary solutions. The argument relies on the concept of local time for semi-martingales (see e.g. [14, Chapter VI.1] for details of the concept). By $[\cdot ]_b$ we denote the quasinorm $[u]_b^2 = \sum _d |u_d|^2 b_d^2.$

Without loss of generality we assume $\tau _0=0$. Otherwise we just need to replace $u( \tau ,x)$ by the process ${\tilde{u}}(\tau ,x):=u(\tau +\tau _0,x)$, apply the lemma with $\tau _0=0$ and with $u_0$ replaced by the initial data ${\tilde{u}}_0^\omega =u^\omega (\tau _0; u_0)$, and then average the estimate in the random ${\tilde{u}}_0^\omega$.

Let us write the solution $u(\tau ;u_0)$ as $u(\tau )=$ $\sum _{d\in {\mathbb {N}}^n}u_d(\tau )\varphi _d$. For any fixed function $g\in C^2({\mathbb {R}})$, consider the process

$$\begin{aligned} f(\tau )=g(\Vert u(\tau \wedge \tau _\Gamma )\Vert _0^2). \end{aligned}$$

Since

$$\begin{aligned} \partial _ug(\Vert u\Vert _0^2)= & {} 2g'(\Vert u\Vert _0^2)\langle \!\langle u,\cdot \rangle \!\rangle , \quad \partial _{uu}g(\Vert u\Vert _0^2)=4g''(\Vert u\Vert _0^2)\langle \!\langle u,\cdot \rangle \!\rangle \langle \!\langle u,\cdot \rangle \!\rangle \\&+2g'(\Vert u\Vert ^2_0)\langle \!\langle \cdot ,\cdot \rangle \!\rangle , \end{aligned}$$

then by Itô’s formula we have

$$\begin{aligned} f(\tau )=f(0)+ \int _0^{\tau \wedge \tau _\Gamma }A(s)ds+\sum _{d\in {\mathbb {N}}^n}b_d\int _0^{\tau \wedge \tau _\Gamma } 2g'(\Vert u(s)\Vert _0^2)\langle u_d(s), d\beta _d(s)\rangle ,\end{aligned}$$

(3.2)

$$\begin{aligned} \begin{aligned} A(s)&=2g'(\Vert u\Vert _0^2)\langle \!\langle u, \Delta u-\frac{1}{\nu }i|u|^2u\rangle \!\rangle +2 \sum _d b_d^2 \big ( g''(\Vert u\Vert _0^2 ) |u_d|^2 +g'(\Vert u\Vert _0^2)\big ) \\&=- 2g'(\Vert u\Vert ^2_0)\Vert u\Vert _1^2+ 2g''(\Vert u\Vert _0^2) [u]_b^2 +2g'(\Vert u\Vert _0^2) B_0, \quad u=u(s). \end{aligned} \end{aligned}$$

(3.3)

Step 1: We firstly show that for any bounded measurable set ${G}\subset {\mathbb {R}}$, denoting by ${\mathbb {I}}_G$ its indicator function, we have the following equality

$$\begin{aligned} \begin{aligned} 2{\mathbb {E}} \int _0^{\tau \wedge \tau _\Gamma }&{\mathbb {I}}_G(f(s)) \,\big (g'(\Vert u(s)\Vert _0^2)\big )^2 \! [u(s)]_b^2 ds =\int _{-\infty }^\infty {\mathbb {I}}_G(a)\\ {}&\Big [{\mathbb {E}}(f(\tau )-a)_+-{\mathbb {E}}(f(0)-a)_+-{\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a+\infty )}(f(s))A(s)ds\Big ]da. \end{aligned}\end{aligned}$$

(3.4)

Let $L(\tau ,a)$, $(\tau ,a)\in [0,\infty )\times {\mathbb {R}}$, be the local time for the semi martingale $f(\tau )$ (see e.g. [14, Chapter VI.1]). Since in view of (3.2) the quadratic variation of the process $f(\tau )$ is

$$\begin{aligned} d \langle f,f \rangle _s = \sum _d (2g'(\Vert u\Vert _0^2) |u_d| b_d)^2 =4\big ( g'(\Vert u\Vert _0^2)\big )^2 [u]_b^2, \end{aligned}$$

then for any bounded measurable set $G\subset {\mathbb {R}}$, we have the following equality (known as the occupation time formula, see [14, Corollary VI.1.6]),

$$\begin{aligned} \int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_G(f(s)) 4\big (g'(\Vert u(s)\Vert _0^2)\big )^2\, [u(s)]_b^2 ds=\int _{-\infty }^\infty {\mathbb {I}}_{{G}}(a)L(\tau ,a)da.\end{aligned}$$

(3.5)

For the local time $L(\tau ,a)$, due to Tanaka’s formula (see [14, Theorem VI.1.2]) we have

$$\begin{aligned} \begin{aligned}(f(\tau )-a)_+=&(f(0)-a)_+\\&+\sum _{d\in {\mathbb {N}}^n} b_d\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,+\infty )}\big (f(s)\big )2g'(\Vert u(s)\Vert _0^2) \langle u_d (s), d\beta _d(s)\rangle \\&+\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,+\infty )}(f(s))A(s)ds+\frac{1}{2}L(\tau ,a). \end{aligned}\end{aligned}$$

(3.6)

Taking expectation of both sides of (3.5) and (3.6) we obtain the required equality (3.4).

Step 2: Let us choose ${G}=[\rho _0,\rho _1]$ with $\rho _1>\rho _0>0$, and $g(x) =g_{\rho _0} (x) \in C^2({\mathbb {R}})$ such that $g'(x)\geqslant 0$, $g(x)=\sqrt{x}$ for $x\geqslant \rho _0$ and $g(x)=0$ for $x\leqslant 0$. Then due to the factors ${\mathbb {I}}_G(f)$ and ${\mathbb {I}}_G(a)$ in (3.4), we may there replace g(x) by $\sqrt{x}$, and accordingly replace $g(\Vert u\Vert ^2_0), g'(\Vert u\Vert ^2_0)$ and $g''(\Vert u\Vert ^2_0)$ by $\Vert u\Vert _0$, $\tfrac{1}{2}\Vert u\Vert _0^{-1}$ and $-\tfrac{1}{4}\Vert u\Vert _0^{-3}$. So the relation (3.4) takes the form

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma } {\mathbb {I}}_G(f(s)) \Vert u(s)\Vert _0^{-2} [u(s)]_b^2 =2\int _{\rho _0}^{\rho _1}\Big [{\mathbb {E}}(f(\tau )-a)_+-{\mathbb {E}}(f(0)-a)_+\Big ]da\\&\quad -2\int _{\rho _0}^{\rho _1}\Big \{{\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,+\infty )}\Big (f(s)\Big )\Big [\frac{2}{2\Vert u(s)\Vert _0}(B_0-\Vert u(s)\Vert _1^2)\\&\quad -\frac{2}{4\Vert u(s)\Vert _0^3} [u(s)]_b^2 \Big ]ds\Big \}da. \end{aligned}\end{aligned}$$

Since the l.h.s. of the above equality is non-negative, we have

$$\begin{aligned} \begin{aligned}&\int _{\rho _0}^{\rho _1}\Big [{\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,+\infty )}\Big (f(s)\Big )\frac{1}{\Vert u(s)\Vert _0^3}\Big (B_0\Vert u(s)\Vert _0^2-\tfrac{1}{2} [u(s)]_b^2 \Big )ds\Big ]da\\&\quad \leqslant \int _{\rho _0}^{\rho _1} {\mathbb {E}} \Big [\big ( (f(\tau )-a)_+ - (f(0)-a)_+\big ) + \int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,+\infty )}\big (f(s)\big )\frac{\Vert u(s)\Vert _1^2}{\Vert u(s)\Vert _0} ds\Big ]da. \end{aligned} \end{aligned}$$

(3.7)

Noting that

$$\begin{aligned} B_0\Vert u\Vert _0^2-\tfrac{1}{2} [u(s)]_b^2 =\sum _{d\in {\mathbb {N}}^n} (B_0-\frac{1}{2}b_d^2)|u_d|^2\geqslant \frac{B_0}{2} \Vert u\Vert _0^2, \end{aligned}$$

that by the definition of the stopping time $\tau _\Gamma$

$$\begin{aligned} (f(\tau )-a)_+-(f(0)-a)_+\leqslant \Gamma , \end{aligned}$$

and that by interpolation,

$$\begin{aligned}\int _0^{\tau \wedge \tau _\Gamma }\frac{\Vert u(s)\Vert _1^2}{\Vert u(s)\Vert _0}ds\leqslant \int _0^{\tau \wedge \tau _\Gamma }\Vert u(s)\Vert _2ds\leqslant (\tau \wedge \tau _\Gamma ) \Gamma , \end{aligned}$$

we derive from (3.7) the relation

$$\begin{aligned} \frac{B_0}{2}\! \int _{\rho _0}^{\rho _1} \big ( {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,+\infty )}\big (f(s)\big ) \Vert u(s)\Vert _0^{-1} ds\big ) da \le (\rho _1-\rho _0) \Gamma \big (1 \,+ \, \tau ) . \end{aligned}$$

When $\rho _0 \rightarrow 0$, we have $g(x) \rightarrow \sqrt{x}$ and $f(\tau ) \rightarrow \Vert u(\tau \wedge \tau _\Gamma )\Vert _0$. So sending $\rho _0$ to 0 and using Fatou’s lemma we get from the last estimate that

$$\begin{aligned} \int _0^{\rho _1}{\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,\infty )}\Big (\Vert u(s)\Vert _0\Big )\Vert u(s)\Vert _0^{-1}dsda\leqslant 2\rho _1(1+\tau )B_0^{-1}\Gamma . \end{aligned}$$

As the l.h.s. above is not smaller than

$$\begin{aligned}\begin{aligned} \frac{1}{\chi }\int _0^{\rho _1}{\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,\chi ]}(\Vert u(s)\Vert _0)dsda, \end{aligned}\end{aligned}$$

then

$$\begin{aligned} \frac{1}{\rho _1}\int _0^{\rho _1} {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,\chi ]}(\Vert u(s)\Vert _0)dsda\leqslant 2(1+\tau )B_0^{-1}\Gamma \,\chi . \end{aligned}$$

(3.8)

By the monotone convergence theorem

$$\begin{aligned} \lim _{a\rightarrow 0} {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(a,\chi ]}(\Vert u(s)\Vert _0)ds = {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(0,\chi ]}(\Vert u(s)\Vert _0)ds, \end{aligned}$$

so we get from (3.8) that

$$\begin{aligned} {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{(0,\chi ]}(\Vert u(s)\Vert _0)ds\leqslant 2(1+\tau )B_0^{-1}\Gamma \chi . \end{aligned}$$

(3.9)

Step 3: We continue to verify that

$$\begin{aligned} {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{\{0\}}(\Vert u(s)\Vert _0)ds=0. \end{aligned}$$

(3.10)

To do this let us fix any index $d\in {\mathbb {N}}^n$ such that $b_d\ne 0$. The process $u_d(\tau )$ is a semimartingale, $\ du_d = v_d ds +b_d d\beta _d, \,$ where $v_d(s)$ is the d-th Fourier coefficient of $\, \Delta u+\frac{1}{\nu }i|u|^2u\,$ for the solution $u(\tau )=\sum _d u_d(\tau ) \varphi _d$ which we discuss. Consider the stopping time

$$\begin{aligned} \tau _R = \inf \{ s\le {\tau \wedge \tau _\Gamma }: |u(s)|_\infty \ge R\}. \end{aligned}$$

Due to (2.3) and (2.6), ${\mathbb {P}} (\tau _R= \tau \wedge \tau _\Gamma ) \rightarrow 1$ as $R\rightarrow \infty$. Let us denote $u_d^R(\tau ) = u_d(\tau \wedge \tau _R).$ To prove (3.10) it suffices to verify that

$$\begin{aligned} \pi (\delta ) := {\mathbb {E}}\int _0^{\tau \wedge \tau _\Gamma }{\mathbb {I}}_{\{ |u_d(s)| <\delta \}} ds \rightarrow 0\quad \text {as} \quad \delta \rightarrow 0. \end{aligned}$$

If we replace above $u_d$ by $u_d^R$, then the obtained new quantity $\pi ^R(\delta )$ differs from $\pi (\delta )$ at most by ${\mathbb {P}} (\tau _R< \tau \wedge \tau _\Gamma )$. The process $u_d^R$ is an Ito process with a bounded drift. So by [7, Theorem 2.2.2, p. 52], $\pi ^R(\delta )$ goes to zero with $\delta$. Thus, given any $\varepsilon >0$, we firstly choose R sufficiently big and then $\delta$ sufficiently small to achieve $\pi (\delta ) <\varepsilon$, for a suitable $\delta (\varepsilon )>0$. So (3.10) is verified. Jointly with (3.9) this proves (3.1). $\square$

4 Lower bounds for Sobolev norms of solutions

In this section we work with Eq. (1.1) in the original time scale t and provide lower bounds for the $H^m$-norms of its solutions with $m>2$. This will prove the assertion (1) of Theorem 1. As always, the constants do not depend on $\nu$, unless otherwise stated.

Theorem 13

For any integer $m\geqslant 3$, if $B_m<\infty$ and

$$\begin{aligned} 0<\kappa <\tfrac{1}{35},\quad T_0\ge 0, \quad T_1>0, \end{aligned}$$

then for any r.v. $u_0(x)\in H^m\cap C_0(K^n)$, satisfying

$$\begin{aligned} {\mathbb {E}}\Vert u_0\Vert _m^2<\infty , \quad {\mathbb {E}}\exp (c\ |u_0|^2_\infty ) \le C <\infty \end{aligned}$$

(4.1)

for some $c, C >0$, we have

$$\begin{aligned} {\mathbb {P}}\left\{ \sup _{T_0 \le t \le T_0+{T_1}{\nu ^{-1}}} \Vert u(t;u_0)\Vert _{m} \ge K \nu ^{-m\kappa } \right\} \rightarrow 1\quad \text {as} \quad \nu \rightarrow 0, \end{aligned}$$

(4.2)

for every $K>0$.

Proof

Consider the complement to the event in (4.2):

$$\begin{aligned} Q^\nu =\left\{ \sup _{T_0\leqslant t \leqslant T_0+\frac{T_1}{\nu }}\Vert u(t )\Vert _{m}<K\nu ^{-m\kappa }\right\} . \end{aligned}$$

We will prove the assertion (4.2) by contradiction. Namely, we assume that there exists a $\gamma >0$ and a sequence $\nu _j\rightarrow 0$ such that

$$\begin{aligned} {\mathbb {P}}( Q^{\nu _j})\geqslant 5\gamma \quad \text {for}\quad j=1,2,\dots , \end{aligned}$$

(4.3)

and will derive a contradiction. Below we write $Q^{\nu _j}$ as Q and always suppose that

$$\begin{aligned} \nu \in \{\nu _1, \nu _2, \dots \}. \end{aligned}$$

The constants in the proof may depend on ${\mathcal {K}}, K, \gamma$, $B_{\,m\vee m_*}$, but not on $\nu$.

Without lost of generality we assume that $T_1=1$. For any $T_0>0$, due to (2.5) and (2.3) the r.v. ${\tilde{u}}_0 := u(T_1)$ satisfies (4.1) with c replaced by c/5. So considering ${\tilde{u}}(t,x) = u(t+T_0,x)$ we may assume that $T_0=0$.

Let us denote $J_1=[0,\frac{1}{\nu }]$. Due to Theorem 7,

$$\begin{aligned} {\mathbb {P}}(Q_1)\geqslant 1-\gamma , \quad Q_1=\{\sup _{t\in J_1}|u(t)|_\infty \leqslant C_1(\gamma )\}, \end{aligned}$$

uniformly in $\nu$, for a suitable $C_1(\gamma )$. Then, by the definition of Q and Sobolev’s interpolation,

$$\begin{aligned} \Vert u^\omega (t)\Vert _l\leqslant C_{l,\gamma }\nu ^{-l\kappa }, \quad \omega \in Q\cap Q_1, \; t\in J_1, \end{aligned}$$

(4.4)

for $l\in [0,m]$ (and any $\nu \in \{\nu _1,\nu _2,\dots \}$).

Denote $J_2=[0,\frac{1}{2\nu }]$ and consider the stopping time

$$\begin{aligned} \tau _1=\inf \{t\in J_2:\Vert u(t)\Vert _2\geqslant C_{2,\gamma }\nu ^{-2\kappa }\} \le \tfrac{1}{2\nu }. \end{aligned}$$

Then $\tau _1=\frac{1}{2\nu }$ for $\omega \in Q\cap Q_1$. So due to (3.1) with $\Gamma =C_{2,\gamma }\nu ^{-2\kappa }$, for any $\chi >0$, we have

$$\begin{aligned} \begin{aligned}{\mathbb {E}}\big (\nu \int _{J_2}{\mathbb {I}}_{[0,\chi ]}(\Vert u(s)\Vert _0)ds{\mathbb {I}}_{Q\cap Q_1}(\omega )\big )&={\mathbb {E}}\big (\nu \int _0^{\frac{1}{2\nu }\wedge \tau _1}{\mathbb {I}}_{[0,\chi ]}(\Vert u(s)\Vert _0)ds{\mathbb {I}}_{Q\cap Q_1}(\omega )\big )\\&\leqslant {\mathbb {E}}\big (\nu \int _0^{\frac{1}{2\nu }\wedge \tau _1}{\mathbb {I}}_{[0,\chi ]}(\Vert u(s)\Vert _0)ds\big )\leqslant C \nu ^{-2\kappa }\chi . \end{aligned} \end{aligned}$$

Consider the event

$$\begin{aligned} \Lambda =\{\omega \in Q\cap Q_1:\Vert u(s)\Vert _0\leqslant \chi , \;\forall s\in J_2\}. \end{aligned}$$

Due to the above, we have,

$$\begin{aligned} {\mathbb {P}}(\Lambda )\leqslant 2 {\mathbb {E}}\big (\nu \int _{J_2}{\mathbb {I}}_{[0,\chi ]}(\Vert u(s)\Vert _0)ds{\mathbb {I}}_{Q\cap Q_1}(\omega )\big )\leqslant 2{C}\nu ^{-2\kappa }\chi . \end{aligned}$$

So ${\mathbb {P}}(\Lambda )\leqslant \gamma$ if we choose

$$\begin{aligned} \chi =c_3(\gamma )\nu ^{2\kappa }, \quad c_3(\gamma )=\gamma (2{C})^{-1}. \end{aligned}$$

(4.5)

Let us set

$$\begin{aligned} Q_2=(Q\cap Q_1)\setminus \Lambda , \quad {\mathbb {P}}(Q_2)\geqslant 3\gamma , \end{aligned}$$

(4.6)

and for $\chi$ as in (4.5), consider the stopping time

$$\begin{aligned} {{\tilde{\tau }}}_1=\inf \{t\in J_2:\Vert u(t)\Vert _0\geqslant \chi \}. \end{aligned}$$

Then ${{\tilde{\tau }}}_1\leqslant \frac{1}{2\nu }$ for all $\omega \in Q_2$. Consider the function

$$\begin{aligned} v(t,x) := u ({{\tilde{\tau }}}_1+t,x), \quad t \in [0,\tfrac{1}{2\nu }]. \end{aligned}$$

It solves Eq. (1.1) with modified Wiener processes and with initial data $v_0(x) = u^\omega ({{\tilde{\tau }}}_1,x)$, satisfying

$$\begin{aligned} \Vert v_0^\omega \Vert _0 \ge \chi =c\nu ^{2\kappa } \quad \text {if}\quad \omega \in Q_2. \end{aligned}$$

(4.7)

Now we introduce another stopping time, in terms of v(t, x):

$$\begin{aligned}\tau _2=\inf \{t\in [0,\tfrac{1}{2\nu }]: \Vert v(t)\Vert _m\geqslant K\nu ^{-m\kappa }\}\le \tfrac{1}{2\nu }. \end{aligned}$$

For $\omega \in Q_2$, $\tau _2=\tfrac{1}{2\nu }$ and in view of (4.4)

$$\begin{aligned} \Vert v^\omega (t)\Vert _l\leqslant C_3(\gamma )\nu ^{-l\kappa }, \quad t\in [0,\tfrac{1}{2\nu }],\; l\in [0,m], \quad \forall \, \omega \in Q_2. \end{aligned}$$

(4.8)

Step 1: Let us estimate from above the increment ${\mathscr {E}}(t,x)=|v(t\wedge \tau _2,x)|^2-|v_0(x)|^2$. Due to Itô’s formula, we have that

$$\begin{aligned} \begin{aligned} {\mathscr {E}}(t,x)&=2\nu \int _{0}^{t\wedge \tau _2}\Big (\langle v(s,x),\Delta v(s,x)\rangle +\sum _{d\in {\mathbb {N}}^n}b_d^2\varphi ^2_d(x)\Big )ds + \sqrt{\nu } \, M(t,x), \\ M(t,x)&= \int _{0}^{t\wedge \tau _2}\sum _{d\in {\mathbb {N}}^n}b_d\varphi _d(x)\langle v(s,x),d\beta _d(s)\rangle . \, \end{aligned} \end{aligned}$$

We treat M as a martingale M(t) in the space $H^1$. Since in view of (A.3) for $0\le s<\tau _2$ we have

$$\begin{aligned} \Vert v(s) \varphi _d\Vert _1 \le C\big ( |v(s)|_\infty \Vert \varphi _d\Vert _1 + |v(s)\Vert _1 |\varphi _d|_\infty \big ) \le C(\zeta d+\zeta ^{(m-1)/m} \nu ^{-\kappa }), \end{aligned}$$

where $\zeta = \sup _{0\leqslant s\leqslant \frac{1}{\nu }}|u(s)|_\infty$ (the assertion is empty if $\tau _2=0$), then for any $0<T_*\le \tfrac{1}{2\nu }$

$$\begin{aligned} {\mathbb {E}}\Vert M(T_*)\Vert _1^2 \le \int _0^{T_*} {\mathbb {E}}\, \sum _d b_d^2 \Vert \varphi _d v(s)\Vert _1^2 ds \le C T_* \nu ^{-2\kappa }, \end{aligned}$$

(4.9)

where we used that $B_1<\infty$. So by Doob’s inequality

$$\begin{aligned} {\mathbb {P}}\Big (\sup _{0\leqslant s\leqslant T_*} \Vert M(s)\Vert _1^2\geqslant r^2\Big )\leqslant C T_*r^{-2} \nu ^{-2\kappa }, \quad \forall r>0. \end{aligned}$$

(4.10)

Let us choose

$$\begin{aligned} T_*=\nu ^{-b}, \quad b\in (0, 1), \end{aligned}$$

where b will be specified later. Then $1\le T_*\le \tfrac{1}{2\nu }$ if $\nu$ is sufficiently small, so due to (4.10)

$$\begin{aligned} {\mathbb {P}}( Q_3)\geqslant 1-\gamma , \quad Q_3=\{\sup _{0\leqslant \tau \leqslant T_*}\Vert M(\tau )\Vert _1\leqslant C_4(\gamma )\nu ^{-\kappa }\sqrt{T_*}\}, \end{aligned}$$

for a suitable $C_4(\gamma )$ (and for $\nu \ll 1$); thus ${\mathbb {P}} (Q_2\cap Q_3) \ge 2\gamma$. Since $\Vert \langle v , \Delta v\rangle \Vert _1 \le C|v|_\infty \Vert v\Vert _3$ by (A.2) and $\Vert \sum _d b_d \varphi _d\Vert _1 \le C,$ then in view of (4.8) and the definition of $Q_3$,

$$\begin{aligned} \Vert {\mathscr {E}}^\omega (\tau )\Vert _1\leqslant C(\gamma )(\nu ^{1-3\kappa }T_*+\nu ^{\frac{1}{2}-\kappa } T_*^{1/2}), \qquad \forall \, \tau \in [0,T_*] , \;\; \forall \,\omega \in Q_2\cap Q_3. \end{aligned}$$

(4.11)

Step 2: For any $x\in K^n$, denoting $R(t) = |v(t,x)|^2, \, a(t) = \Delta v(t,x)$ and $\xi (t) = \xi (t,x)$, we write the equation for $v(t) := v(t,x)$ as an Itô process:

$$\begin{aligned} dv(t) = (-i R v +\nu a) dt + \sqrt{\nu }\, d\xi (t). \end{aligned}$$

(4.12)

Setting $w(t) = e^{i\int _0^t R(s)ds} v(t)$, we observe that w also is an Itô process, $w(0) =v_0$ and $dv = e^{-i\int _0^t R(s)ds} dw -i Rv\, dt.$ From here and (4.12),

$$\begin{aligned} w(t) = v_0 +\nu \int _0^t e^{i\int _0^s R(s')ds'} a(s) ds + \sqrt{\nu }\, \int _0^t e^{i\int _0^s R(s')ds'} d \xi (s). \end{aligned}$$

So $v(t\wedge \tau _2)= v(t\wedge \tau _2,x)$ can be written as

$$\begin{aligned} v(t\wedge \tau _2,x)= I_1(t\wedge \tau _2,x) + I_2(t\wedge \tau _2,x)+ I_3(t\wedge \tau _2,x), \end{aligned}$$

(4.13)

where

$$\begin{aligned}\begin{aligned}&I_1(t,x)= e^{-i\int _0^{t}|v(s,x)|^2ds}v_0,\quad I_2(t,x)=\nu \int _0^{t}e^{-i\int _{s}^{t}|v(s',x)|^2ds'}\Delta v(s,x)ds, \\&I_3(t,x)= \sqrt{\nu } e^{-i\int _0^{t}|v(s',x)|^2ds'} \int _0^{t}e^{i\int _0^{s}|v(s',x)|^2ds'}d\xi (s,x) . \end{aligned} \end{aligned}$$

Our next goal is to obtain a lower bound for $\Vert v(T_*)\Vert _1$ when $\omega \in Q_2\cap Q_3$, using the above decomposition (4.13).

Step 3: We first deal with the stochastic term $I_3(t)$. For $0 \leqslant s\leqslant s_1\leqslant T_*\wedge \tau _2$ we set

$$\begin{aligned} \begin{aligned} W(s,s_1, x):= \exp ( i \int _{s}^{s_1}|v(s',x)|^2ds'), \quad F(s,s_1,x) := \int _{s}^{s_1}|v(s',x)|^2ds'; \end{aligned} \end{aligned}$$

(4.14)

then $W(s,s_1, x) = \exp \big (i F(s,s_1, x)\big )$. The functions F and W are periodic in x, but not odd. Speaking about them we understand $\Vert \cdot \Vert _m$ as the non-homogeneous Sobolev norm, so $\Vert F\Vert _m^2 = \Vert F\Vert _0^2 + \Vert ( -\Delta )^{m/2} F\Vert _0^2,$ etc. We write $I_3$ as

$$\begin{aligned} I_3(t) =\sqrt{\nu }\ {\overline{W}} ( 0,t \wedge \tau _2 ,x) \int _0^{t\wedge \tau _2} W(0,s,x) d\xi (s,x). \end{aligned}$$

(4.15)

In view of (A.1),

$$\begin{aligned} \Vert \exp (iF(s,s_1 \cdot ))\Vert _{k} \le C _{k}(1+ |F(s,s_1, \cdot )|_\infty )^{k-1} \Vert F(s,s_1,\cdot )\Vert _{k}, \quad k \in {\mathbb {N}}. \end{aligned}$$

(4.16)

For any $s\in J =[ 0, T_*\wedge \tau _2)$, by (A.3) and the definition of $\tau _2$, we have that $v:=v(s)$ satisfies

$$\begin{aligned} \Vert |v|^2\Vert _{1} \le C |v|_\infty \Vert v\Vert _{1} \le C |v|_\infty \Vert v\Vert _0^{1- 1/m} \Vert v\Vert _m^{1/m} \le C' |v|_\infty ^{2-1/m} \nu ^{-\kappa } \end{aligned}$$

(4.17)

(this assertion is empty if $\tau _2=0$ since then $J=\emptyset$). So for $s,s_1 \in J$,

$$\begin{aligned}&|F(s,s_1,\cdot )|_\infty \leqslant |s_1-s| \sup _{s'\in J}|v(s')|^2_\infty ,\quad \Vert F (s,s_1,\cdot )\Vert _{k} \\&\quad \le C \nu ^{ -\kappa k} |s_1-s|\big (\sup _{s' \in J} | v(s')|_\infty \big )^{2-k/m}\;\; \end{aligned}$$

for $k\le m$. Then, due to (4.16),

$$\begin{aligned} \Vert W(0,s\wedge \tau _2,\cdot )\Vert _{1} \leqslant C' T_* \nu ^{-\kappa } (1+ \sup _{s \in J} |v(s)|_\infty ^{2} ). \end{aligned}$$

(4.18)

Consider the stochastic integral in (4.15),

$$\begin{aligned} N(t,x)=\int _0^{t}W(0,s,x) d\xi (s,x). \end{aligned}$$

The process $t \mapsto W(0,t,x)$ is adapted to the filtration $\{ {\mathcal {F}}_t\}$, and

$$\begin{aligned}dW(0,t,x)=i|v(t,x)|^2W(0,t,x)dt.\end{aligned}$$

So integrating by parts (see, e.g., [14, Proposition IV.3.1]) we re-write N as

$$\begin{aligned}N(t,x)=W(0,t,x)\xi (t,x)- i \int _0^t\xi (s,x) |v(s,x)|^2W(0,s,x)ds, \end{aligned}$$

and we see from (4.15) that

$$\begin{aligned} I_3(t)=\sqrt{\nu }\xi (t\wedge \tau _2,x)+ i\sqrt{\nu }\int _0^{t\wedge \tau _2}\xi (s,x) |v(s,x)|^2W(s,t\wedge \tau _2,x)ds.\nonumber \\ \end{aligned}$$

(4.19)

Due to (1.4) and since $B_m<\infty$, the Wiener process $\xi (t,x)$ satisfies

$$\begin{aligned}{\mathbb {E}}\Vert \xi (T_*,x)\Vert _1^2\leqslant CB_1T_*,\end{aligned}$$

and

$$\begin{aligned}{\mathbb {E}}\sup _{0\leqslant t\leqslant T_*}|\xi (t,\cdot )|_\infty \leqslant \sum _{d\in {\mathbb {N}}^n}b_d({\mathbb {E}}\sup _{0\leqslant t\leqslant T_*}|\beta _d(t)\varphi _d|_\infty )\leqslant CB_*\sqrt{T_*},\end{aligned}$$

(we recall that $B_*=\sum _{d\in {\mathbb {N}}^n}|b_d|<\infty$). Therefore,

$$\begin{aligned} {\mathbb {P}}(Q_4)\geqslant 1-\gamma , \quad Q_4=\{ \sup _{0\leqslant t\leqslant T_*} ( \Vert \xi (t)\Vert _1\vee |\xi (t)|_\infty ) \leqslant CT_*^{1/2}\}, \end{aligned}$$

with a suitable $C=C(\gamma )$. Let

$$\begin{aligned} {\tilde{Q}}=\bigcap _{i=1}^4 Q_i, \end{aligned}$$

then ${\mathbb {P}}({\tilde{Q}})\geqslant \gamma$. As $\tau _2 = T_*$ for $\omega \in {\tilde{Q}}$, then due to (4.17), (4.18), (4.19) and (A.3), for $\omega \in {\tilde{Q}}$ we have

$$\begin{aligned} \begin{aligned} \sup _{0\leqslant t\leqslant T_*} \Vert I^\omega _3(t)\Vert _1&\le \sqrt{\nu }\, \sup _{0\leqslant t\leqslant T_*} \Big (\Vert \xi ^\omega (t)\Vert _1+\int _0^t\Vert \xi ^\omega (s)|v^\omega (s)|^2W^\omega (s,t)\Vert _1ds\Big )\\&\leqslant {C} T_*^{5/2}\nu ^{\frac{1}{2}-\kappa }. \end{aligned} \end{aligned}$$

(4.20)

Setp 4: We then consider the term $I_{2}= \nu \int _0^{t\wedge \tau _2} {\bar{W}}(s, t\wedge \tau _2,x) \Delta v(s,x)ds$. To bound its $H^1$-norm we need to estimate $\Vert W\Delta v\Vert _1$. Since

$$\begin{aligned} \Vert \partial _x^a W\partial _x^bv\Vert _0\leqslant C \Vert W\Vert _3^{1/3}\Vert v\Vert _3^{2/3}|v|_\infty ^{1/3}\quad \text {if} \;\; |a|=1,|b|=2, \end{aligned}$$

(see [17, Proposition 3.6]), we have

$$\begin{aligned} \Vert W\Delta v\Vert _1\leqslant C(\Vert v\Vert _3+ \Vert W\Vert _3^{1/3}\Vert v\Vert _3^{2/3}|v|_\infty ^{1/3}).\end{aligned}$$

Then in view of (4.16) and (4.8), for $\omega \in {\tilde{Q}}$

$$\begin{aligned} \Vert W\Delta v\Vert _1 \le C\big ( \nu ^{-3\kappa } +(T_*^3 \nu ^{-3\kappa })^{1/3} \nu ^{-2\kappa } \big ) \le C \nu ^{-3\kappa } T_*, \end{aligned}$$

and accordingly

$$\begin{aligned} \sup _{0\leqslant t\leqslant T_*}\Vert I^\omega _2(t)\Vert _1\leqslant \nu \sup _{0\leqslant t\leqslant T_*}\!\int _0^{t}\Vert W^\omega (s,T_*)\Delta v^\omega (s)\Vert _1ds \leqslant C \nu ^{1-3\kappa }T_*^{2},\;\; \; \forall \,\omega \in {\tilde{Q}}. \end{aligned}$$

(4.21)

Step 5: Now we estimate from below the $H^1$-norm of the term $I^\omega _{1}(T_*,x)$, $\omega \in {\tilde{Q}}$. Writing it as $I^\omega _1(T_*,x) = e^{-i T_*|v_0(x)|^2} e^{ -i \int _0^{T_*} {\mathscr {E}} (s,x) ds} v_0(x)$ wee see that

$$\begin{aligned} \Vert I^\omega _1(T_*) \Vert _1\geqslant \Vert \nabla (\exp (-iT_*|v_0|^2)v_0\Vert _0-\Vert \nabla (\exp (-i\int _0^{T_*} {\mathscr {E}} (s)ds))v_0\Vert _0-\Vert v_0\Vert _1. \end{aligned}$$

This first term on the r.h.s is

$$\begin{aligned} T_*\Vert v_0\nabla (|v_0|^2)\Vert _0=T_*\tfrac{2}{3}\Vert \nabla |v_0|^3\Vert _0\geqslant CT_* \Vert |v_0|^3\Vert _0\geqslant CT_*\Vert v_0\Vert _0^3\ge C T_* \nu ^{6\kappa },\quad C>0, \end{aligned}$$

where we have used the fact that $u|_{\partial K^{n}}=0$, Poincaré’s inequality and (4.7).

For $\omega \in {\tilde{Q}}$ and $0\le s\le T_*$, in view of (4.11), the second term is bounded by

$$\begin{aligned} \begin{aligned} \left\| \left( \int _0^{T_*}\nabla {\mathscr {E}}(s)ds\right) v_0\right\| _0 \leqslant CT_*|v_0|_\infty \sup _{0\leqslant s\leqslant T_*}\Vert {\mathscr {E}}(s)\Vert _1 \le {C} T_*( \nu ^{1-3\kappa } T_* + \nu ^{\frac{1}{2}-\kappa } T_*^{1/2} ). \end{aligned} \end{aligned}$$

Therefore, using (4.11), we get for the term $I^\omega _1(T_*)$ the following lower bound:

$$\begin{aligned} \Vert I_{1}^\omega (T_*)\Vert _1\geqslant C \Big (\nu ^{6\kappa }T_*- T_*\big (\nu ^{1-3\kappa }T_* +\nu ^{\frac{1}{2}-\kappa } T_*^{1/2}\big ) -\nu ^{-\kappa } \Big ). \end{aligned}$$

Recalling $T_*=\nu ^{-b}$ we see that if we assume that

$$\begin{aligned} {\left\{ \begin{array}{ll} 6\kappa -b<-\kappa ,\\ 6\kappa -b< 1-3\kappa -2b, \\ 6\kappa -b < 1/2 -\kappa -\frac{3}{2}b , \end{array}\right. } \end{aligned}$$

(4.22)

then for $\omega \in {\tilde{Q}}$,

$$\begin{aligned} \Vert I_{1}^\omega (T_*)\Vert _1\geqslant C \nu ^{6\kappa }T_*, \quad C >0, \end{aligned}$$

(4.23)

provided that $\nu$ is sufficiently small.

Step 6: Finally, remembering that $\tau _2=T_*$ for $\omega \in {\tilde{Q}}$ and combining the relations (4.20), (4.21) and (4.23) to estimate the terms of (4.13), we see that for $\omega \in {\tilde{Q}}$ we have

$$\begin{aligned} \Vert v^\omega (T_*)\Vert _1\geqslant \Vert I_{1}^\omega (T_*)\Vert _1-\Vert I^\omega _2(T_*)\Vert _1-\Vert I^\omega _3(\tau _*)\Vert _1\geqslant \tfrac{1}{2}C_1 \nu ^{6\kappa -b}, \quad C_1>0, \end{aligned}$$

(4.24)

if we assume in addition to (4.22) that

$$\begin{aligned} 6\kappa -b<\frac{1}{2}-\kappa -\frac{5}{2}b, \end{aligned}$$

(4.25)

and $\nu$ is small. Note that this relation implies the last two in (4.22).

Combining (4.8) and (4.24) we get that

$$\begin{aligned} \nu ^{-b+7\kappa }\leqslant C^{-1}_2, \end{aligned}$$

(4.26)

for all sufficiently small $\nu$. Thus we have obtained a contradiction with the existence of the sets $Q^{\nu _j}$ as at the beginning of the proof if (for a chosen $\kappa$) we can find a $b\in (0,1)$ which meets (4.22), (4.25) and

$$\begin{aligned} -b+7\kappa <0. \end{aligned}$$

Noting that this is nothing but the first relation in (4.22), we see that we have obtained a contradiction if

$$\begin{aligned} \kappa< \tfrac{1}{7} b, \quad \kappa < \tfrac{1}{14} -\tfrac{3}{14} b, \end{aligned}$$

for some $b\in (0,1)$. We see immediately that such a b exists if and only if $\kappa < \tfrac{1}{35}$.

$\square$

Amplification If we replace the condition $m\geqslant 3$ with the weaker assumption

$$\begin{aligned} {\mathbb {R}}\ni m>2, \end{aligned}$$

then the statement (4.2) remains true for $0<\kappa <\kappa (n,m)$ with a suitable (less explicit) constant $\kappa (n,m)>0$. In this case we obtain a contradiction with the assumption (4.3) by deriving a lower bound for $\Vert v(T_*)\Vert _\alpha$, where $\alpha =\min \{1,m-2\}\in (0,1]$, using the decomposition (4.13). The proof remains almost identical except that now, firstly, we bound $\Vert I_{2}\Vert _\alpha$ ($\alpha <1$) from above using the following estimate from [15, Theorem 5, p. 206] (also see there p. 14):

$$\begin{aligned}\Vert W\Delta u\Vert _\alpha \leqslant C\Vert u\Vert _{2+\alpha }(|W|_\infty +|W|_\infty ^{1-\frac{2\alpha }{n}}\Vert W\Vert _2^{\frac{2\alpha }{n}});\end{aligned}$$

and, secondly, estimate $\Vert I_{1}^\omega (T_*)\Vert _\alpha$ ($\alpha <1$) from below as

$$\begin{aligned} \Vert I_{1}^\omega (T_*)\Vert _\alpha \geqslant {\Vert I_{1}^\omega (T_*)\Vert _1^{2-\alpha }} \,{\Vert I^\omega _2(T_*)\Vert _2^{-1+\alpha }}, \end{aligned}$$

which directly follows from Sobolev’s interpolation. See [6] for more details.

5 Lower bounds for time-averaged Sobolev norms

In this section we prove the assertion (2) of Theorem 1. We provide each space $H^r$, $r\ge 0$, with the scalar product

$$\begin{aligned} \langle \!\langle u,v\rangle \!\rangle _r:=\langle \!\langle (-\Delta )^{\frac{r}{2}}u,(-\Delta )^{\frac{r}{2}}v\rangle \!\rangle , \end{aligned}$$

corresponding to the norm $\Vert u\Vert _r$. Let $u(t) =\sum u_d(t) \varphi _d$ be a solution of Eq. (1.1). Applying Itô’s formula to the functional $\Vert u\Vert _m^2$, we have for any $0\leqslant t<t'<\infty$ the relation

$$\begin{aligned} \begin{aligned} \Vert u(t')\Vert _m^2=&\Vert u(t)\Vert _m^2+2\int _{t}^{t'}\langle \!\langle u(s), \nu \Delta u(s) - i|u(s)|^2u(s)\rangle \!\rangle _mds\\&+2\nu B_m(t'-t)+2\sqrt{\nu } M(t,t'), \end{aligned}\end{aligned}$$

(5.1)

where M stands for the real scalar productis the stochastic integral

$$\begin{aligned} M(t,t'):= \int _t^{t'}\sum _{d\in {\mathbb {N}}^n} b_d|d|^{2m} \langle u_d(s), d\beta _d(s)\rangle . \end{aligned}$$

Let us fix a $\gamma \in (0,\frac{1}{8})$. Due to Theorems 7 and  13, for small enough $\nu$ there exists an event $\Omega _1\subset \Omega$, ${\mathbb {P}}(\Omega _1)\geqslant 1-\gamma /2,$ such that for all $\omega \in \Omega _1$ we have:

• a) $\sup _{0\leqslant t\leqslant \frac{1}{\nu }}|u^\omega (t)|_\infty \leqslant C(\gamma )$, for a suitable $C(\gamma )>0$;

• b) there exist $t_\omega \in [0,\frac{1}{3\nu }]$ and $t_\omega '\in [\frac{2}{3\nu },\frac{1}{\nu }]$ satisfying

  $$\begin{aligned} \Vert u^\omega (t_\omega )\Vert _m,\;\Vert u^\omega (t_\omega ')\Vert _m\geqslant \nu ^{-m\kappa }. \end{aligned}$$

  (5.2)

Since for the martingale M(0, t) we have that

$$\begin{aligned} {\mathbb {E}}|M(0,\tfrac{1}{\nu })|^2\leqslant B_m {\mathbb {E}}\int _0^\frac{1}{\nu } \Vert u(s)\Vert _m^2ds =: X_m, \end{aligned}$$

then by Doob’s inequality

$$\begin{aligned} {\mathbb {P}}(\Omega _2)\geqslant 1-\frac{\gamma }{2}, \qquad \Omega _2=\Big \{ \sup _{0\leqslant t\leqslant \frac{1}{\nu }}|M(0,t)|\leqslant c(\gamma ) X_m^{1/2} \Big \}. \end{aligned}$$

Now let us set $\ {\hat{\Omega }}=\Omega _1\cap \Omega _2.$ Then ${\mathbb {P}}({\hat{\Omega }})\geqslant 1-\gamma$ for small enough $\nu$, and for any $\omega \in {\hat{\Omega }}$ there are two alternatives:

(i) there exists a $t_\omega ^0\in [0,\frac{1}{3\nu }]$ such that $\Vert u^\omega (t^0_\omega )\Vert _m=\frac{1}{3}\nu ^{-\kappa m}$. Then from (5.1) and (5.2) in view of (A.4) we get

$$\begin{aligned}\begin{aligned}&\frac{8}{9}\nu ^{-2m\kappa }+2\nu \int _{t_\omega ^0}^{t'_\omega }\Vert u^\omega (s)\Vert _{m+1}^2ds \leqslant C(m,\gamma ) \int _0^{\frac{1}{\nu }} \Vert u^\omega (s)\Vert _m^2ds+2B_m\\&\quad +2\sqrt{\nu }c(\gamma ) X_m^{1/2}. \end{aligned} \end{aligned}$$

(ii) There exists no $t\in [0,\frac{1}{3\nu }]$ with $\Vert u^\omega (t)\Vert _m=\frac{1}{3}\nu ^{-\kappa m}$. In this case, since $\Vert u^\omega (t)\Vert _m$ is continuous with respect to t, then due to (5.2) $\Vert u^\omega (t)\Vert _m > \frac{1}{3}\nu ^{-m\kappa }$ for all $t\in [0,\frac{1}{3\nu }]$. This leads to the relation

$$\begin{aligned} \tfrac{1}{27}\nu ^{-2m\kappa -1}\leqslant \int _0^{\frac{1}{\nu }}\Vert u^\omega (s)\Vert _m^2ds. \end{aligned}$$

In both cases for $\omega \in {\hat{\Omega }}$ we have:

$$\begin{aligned}\frac{1}{27}\nu ^{-2m\kappa }\leqslant C'(m,\gamma ) \int _0^{\frac{1}{\nu }} \Vert u(s)\Vert _m^2ds+2B_m +{\nu } c(\gamma )^2 + X_m. \end{aligned}$$

It implies that

$$\begin{aligned} {\mathbb {E}}\nu \int _0^{\frac{1}{\nu }}\Vert u(\tau )\Vert _m^2d\tau \geqslant C\nu ^{-2m\kappa +1} \end{aligned}$$

(for small enough $\nu$), and gives the lower bound in (1.7).

The upper bound follows directly from Theorem 8.

Proof of Corollaries 3 and 4

Since $B_k<\infty$ for each k and all coefficients $b_d$ are non-zero, then Eq. (1.1) is mixing in the spaces $H^m$, $m\in {\mathbb {N}}$, see Corollary 10. As the stationary solution $v^{st}$ satisfies Corollary 11 with any m, then for each $\mu \in {\mathbb {N}}$ and $M>0$, interpolating the norm $\Vert u\Vert _\mu$ via $\Vert u\Vert _0$ and $\Vert u\Vert _m$ with m sufficiently large we get that the stationary measure $\mu _\nu$ satisfies

$$\begin{aligned} \int \Vert u\Vert _\mu ^M \mu _\nu (du) < \infty \quad \forall \, \mu \in {\mathbb {N}}, \;\forall M>0. \end{aligned}$$

(5.3)

Similar, in view of (2.5) and Theorem  7,

$$\begin{aligned} {\mathbb {E}}\Vert u(t;u_0)\Vert _\mu ^M \le C_\nu (u_0) \quad \forall t\ge 0, \end{aligned}$$

(5.4)

for each $u_0\in C^\infty$ and every $\mu$ and M as in (5.3). Now let us consider the integral in (1.7) and write it as

$$\begin{aligned} J_t := \nu \int _t^{t+\nu ^{-1}} {\mathbb {E}}\Vert u(s)\Vert _m^2ds. \end{aligned}$$

Replacing the integrand in $J_t$ with ${\mathbb {E}}( \Vert u_\nu (s)\Vert _m \wedge N)^2$, $N\ge 1$, using the convergence

$$\begin{aligned} {\mathbb {E}}\big ( \Vert u(s;v_0) \Vert _m \wedge N\big )^2 \rightarrow \int \big ( \Vert u \Vert _m \wedge N\big )^2 \mu _\nu (du) \quad \text {as} \quad s\rightarrow \infty \quad \forall \,N, \end{aligned}$$

(5.5)

which follows from Corollary 10, and the estimates (5.3), (5.4) we get that

$$\begin{aligned} J_t \, \rightarrow \, \int \Vert u\Vert _m^2 \,\mu _\nu (du)\quad \text {as} \quad t\rightarrow \infty . \end{aligned}$$

(5.6)

This convergence and (1.7) imply the assertion of Corollary 3.

Now the convergence (5.5) jointly with estimates (5.3), (5.4) and (1.8) imply Corollary 4. $\square$

Appendices

Appendix A. Some estimates

For any integer $l\in {\mathbb {N}}$ and $F\in H^l$ we have that

$$\begin{aligned} \Vert \exp (iF(x))\Vert _{l} \le C_{l} (1+ |F|_\infty )^{l-1} \Vert F\Vert _{l}. \end{aligned}$$

(A.1)

Indeed, to verify (A.1) it suffices to check that for any non-zero multi-indices $\beta _1,\dots ,\beta _{l'}$, where $1\le l'\leqslant l$ and $|\beta _1| +\cdots + |\beta _{l'}| =l,$ we have

$$\begin{aligned} \Vert \partial _x^{\beta _1} F\cdots \partial _x^{\beta _{l'}} F \Vert _0 \le C |F|_\infty ^{l'-1} \Vert F\Vert _l. \end{aligned}$$

(A.2)

But this is the assertion of Lemma 3.10 in [17]. Similarly,

$$\begin{aligned} \Vert F G \Vert _r \leqslant C_r ( |F|_\infty \Vert G\Vert _r + |G|_\infty \Vert F\Vert _r), \quad F,G \in H^r,\;\; r\in {\mathbb {N}}, \end{aligned}$$

(A.3)

see [17, Proposition 3.7] (this relation is known as Moser’s estimate). Finally, since for $| \beta | \le m$ we have $| \partial _x^\beta v|_{2m/ \beta |} \le C |v|_\infty ^{1- |\beta | /m} \Vert v\Vert _m^{|\beta | /m}$ (see relation (3.17) in [17]), then

$$\begin{aligned} |\langle \!\langle |v|^2 v, v\rangle \!\rangle _m| \le C_m \Vert v\Vert _m^2 |v|_\infty ^2, \qquad |\langle \!\langle |v|^2 v, v\rangle \!\rangle _m| \le C'_m \Vert v\Vert _{m+1}^{\frac{2m}{m+1}} |v|_\infty ^{\frac{2m+4}{m+1}} . \end{aligned}$$

(A.4)

Appendix B. Proof of Theorem 8

Applying Ito’s formula to a solution $v(\tau )$ of Eq. (2.1) we get a slow time version of the relation (5.1):

$$\begin{aligned} \begin{aligned} \Vert v(\tau )\Vert _m^2=\Vert v_0\Vert _m^2+2 \int _{0}^{\tau } \big ( -\Vert v \Vert ^2_{m+1} - \nu ^{-1} \langle \!\langle i|v |^2v , v \rangle \!\rangle _m \big )ds +2 B_m \tau +2 M(\tau ), \end{aligned} \end{aligned}$$

(B.1)

where $M(\tau )= \int _0^{\tau }\sum _{d} b_d|d|^{2m} \langle v_d(s), d\beta _d(s)\rangle .$ Since in view of (A.4)

$$\begin{aligned} {\mathbb {E}}\big | \langle \!\langle |v|^2v, v\rangle \!\rangle _m \big | \le C_m \big ( {\mathbb {E}}\Vert v\Vert _{m+1}^{2}\big )^{\frac{m}{m+1}} {\mathbb {E}}\big ( |v|_\infty ^{2m+4}\big )^{\frac{1}{m+1}}, \end{aligned}$$

then denoting ${\mathbb {E}}\Vert v(\tau )\Vert _r^2 =: g_r(\tau ), \ r \in {\mathbb {N}}\cup \{0\},$ taking the expectation of (B.1), differentiating the result and using (2.3), we get that

$$\begin{aligned} \frac{d}{d\tau } g_m \le -2 g_{m+1} +C_m \nu ^{-1} g_{m+1}^{\frac{m}{m+1}} +2B_m \le -2 g_{m+1}\big ( 1- C'_m \nu ^{-1} g_{m}^{-\frac{1}{m}} +2B_m\big ), \end{aligned}$$

(B.2)

since $g_m \le g_0^{1/(m+1)} g_{m+1}^{m/(m+1)} \le C_m g_{m+1}^{m/(m+1)} .$ We see that if $g_m \ge (2\nu ^{-1} C'_m)^m$, then the r.h.s. of (B.2) is

$$\begin{aligned} \le -g_{m+1} +2B_m \le -C_m^{-1} g_m^{(m+1)/m} +2B_m \le - {\bar{C}}_m \nu ^{-m-1} + 2B_m, \end{aligned}$$

(B.3)

which is negative if $\nu \ll 1$. So if

$$\begin{aligned} g_m(\tau ) < (2\nu ^{-1} C'_m)^m \end{aligned}$$

(B.4)

at $\tau =0$, then (B.4) holds for all $\tau \ge 0$ and (2.4) follows. If $g_m(0)$ violates (B.4), then in view of (B.2) and (B.3), for $\tau \ge 0$, while (B.4) is false, we have that

$$\begin{aligned} \frac{d}{d\tau } g_m \le -C_m g_{m}^{(m+1)/m} +2B_m, \end{aligned}$$

which again implies (2.4). Besides, in view of (B.2),

$$\begin{aligned} \frac{d}{d\tau } g_m \le - g_{m} +C_m (\nu , |v_0|_\infty , B_{m_*},B_m). \end{aligned}$$

This relation immediately implies (2.5).

Now let us return to Eq. (B.1). Using Doob’s inequality and (2.4) we find that

$$\begin{aligned} {\mathbb {E}}(\sup _{0\le \tau \le T} |M(\tau )|^2 ) \le C<\infty . \end{aligned}$$

Next, applying (A.4) and Young’s inequality we get

$$\begin{aligned} \int _{0}^{\tau } \big ( -\Vert v \Vert ^2_{m+1} - \nu ^{-1} \langle \!\langle i|v |^2v , v \rangle \!\rangle _m \big )ds \le C_m \int _{0}^{\tau } | v(s)|_\infty ^{2m+3} ds, \quad \forall \ 0\le \tau \le T. \end{aligned}$$

Finally, using in (B.1) the last two displayed formulas jointly with (2.3) we obtain (2.6).