Abstract
This paper develops an averaging approach on macroscopic scales to derive Smoluchowski–Kramers approximation for a Langevin equation with state dependent friction in d-dimensional space. In this approach we couple the microscopic dynamics to the macroscopic scales. The weak convergence rate is also presented.
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1 Introduction
The Smoluchowski–Kramers (SK) approximation is useful to describe the motion of a particle with small mass which has been studied in lots of works beginning with Smoluchowski [20] and Kramers [17]. The motion of a particle with mass \(0< \epsilon \ll 1\) in \({{\mathbb {R}}}^{d}\) (\(d\ge 1\)) is described by the following Langevin equation
where constant friction \(\alpha >0\) , \(F(x): {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d}\), \(\sigma (x): {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d\times k}\) and \(\{B_t\}\) is k-dimensional standard Wiener process. The classical SK approximation states that for every \(T>0\)
with
For more detail one can refer to [8] . The above limit equation, with just letting \(\epsilon =0\) , is not surprising for such constant friction \(\alpha \). However a noise-induced drift was observed in experiment [21] for the case of state dependent friction, which implies the limit equation can not be obtained by letting \(\epsilon =0\) . Recent work by Hottovy et al. [14] presented a mathematical explanation, but lack of some intuition, by a theory of the convergence of stochastic integral with respect to semimartingale, for such experimental observation.
In this paper we present a new approach which makes the limit equation more intuitively, although in a weak sense. We consider the following Langevin equation with state dependent friction,
where \(\alpha (x): {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d\times d}\) is a \(d\times d\) invertible matrix-valued function. Our idea is to consider the limit of \(\rho _t^\epsilon \), the law of \(x^\epsilon _t\) , as \(\epsilon \rightarrow 0\) . For this we first write out the equations solved by \(\rho _t^\epsilon \) (see (1.6)–(1.7)). However, these equations are not closed, we couple the equations (1.1)–(1.2) to (1.6)–(1.7) . Then we pass the limit \(\epsilon \rightarrow 0\) in equations (1.6)–(1.7) via an averaging approach.
Typically, write the equation (1.1) into the following equivalent form
First it is known that the law \(f_t^\epsilon \in {\mathcal {P}}({\mathbb {R}}^d\times {\mathbb {R}}^d)\), the set consisting of all probability measures on \({{\mathbb {R}}}^{d}\times {{\mathbb {R}}}^{d}\) , of \(({x}^{\epsilon }_t,{\dot{x}}^{\epsilon }_t)\) satisfies the Fokker–Planck equation
in the weak sense, where \(a(x)=\sigma (x)\sigma ^\top (x)\) and \(\sigma ^\top (x)\) is the transpose of \(\sigma (x)\), that is for \(\varphi \in C_0^\infty ({\mathbb {R}}^d\times {\mathbb {R}}^d)\),
The law of \(x^\epsilon _t\) is
and define
Integrating both sides of the equation (1.5) with respect to v, and multiplying both sides of (1.5) by v , then integrating with respect to v, we get
Notice that
providing \(\rho _{t}^{\epsilon }\ne 0\) . Here \({\mathbb {E}}^x(v_t^\epsilon \otimes v_t^\epsilon )\) is the expectation with fixing \(x^\epsilon =x\) in equation (1.4) . Thus, we obtain the following closed system
The above equations (1.8) is in the form of a slow-fast system with slow component \((\rho _t^\epsilon , x^{\epsilon }_{t})\) and fast component \((Y_t^\epsilon , v^{\epsilon }_{t})\). So an averaging method is applicable to pass the limit \(\epsilon \rightarrow 0\) [7, 13, 18, 19, e.g.]. In fact, by the idea of averaging approach [7, Chapter 5], fixing \(\rho ^{\epsilon }_{t}\) to a probability measure \( \rho \) (see equation (4.1) and Remark 4.1), we have \(Y^\epsilon _t\), as \(\epsilon \rightarrow 0\) , converges weakly to (see Lemma 4.1)
Then we formally derive the limit equation (2.4) by replacing \(Y^\epsilon _t\) by \(Y^{*,\rho }\) in the first equation of (1.8). We call the above an averaging principle on macroscopic scale.
There are a lot of literature about SK approximation in case of variable friction. Freidlin and Hu [9] considered the SK approximation for (1.1) by regularizing the noise. Freidlin, Hu and Wentzell [10], also by regularization method, considered the SK approximation with some degenerating friction. There are also some works on SK approximation of infinite dimensional system with constant damping [3] and state–dependent damping [5, 6, e.g.] and some related problem, large deviation e.g. [4].
The rest of this paper is organized as follows. In Sect. 2, we give some preliminaries, assumptions and the main result. The tightness of \(\{\rho ^{\epsilon }_{t}\}\) is shown in Sect. 3, then the averaging procedure is implemented in the last section. It should be clarified that the positive constant C and \(C_T\) may be different from line to line in the proofs.
2 Preliminaries and Main Result
Let \((\Omega ,{\mathcal {F}},{\mathbb {P}})\) be a complete probability space, and \({\mathbb {E}}\) denote the expectation with respect to \({\mathbb {P}}\) . Denote by \(|\cdot |\) the norm on \({\mathbb {R}}^d\) and \(\langle \cdot ,\cdot \rangle \) the inner product in space \(L^2({{\mathbb {R}}}^{d})\) .
We make the following assumptions.
\((\mathbf {H_1})\) \(\alpha (x): {\mathbb {R}}^d\rightarrow {\mathbb {R}}^{d\times d}\) is continuous differentiable function. The smallest eigenvalue \(\lambda _1(x)\) of \(\frac{1}{2}(\alpha +\alpha ^{\top })\) is positive uniformly with respect to x, i.e. for some constant \(C_{\lambda _\alpha }>0\) ,
\((\mathbf {H_2})\) F(x) and \(\sigma (x)\) are continuous differentiable and Lipschitz functions with Lipschitz constant \(C_F\) and \(C_{\sigma }\) respectively, i.e., for \(x,y\in {\mathbb {R}}^d\),
\((\mathbf {H_3})\) There is a constant \(C>0\) such that \(|\partial _{x_{k}}\alpha _{ij}(x)|\le C\), for all \(1\le i,j, k \le d\), \(x\in {\mathbb {R}}^d\) .
Remark 2.1
Since the global Lipschitz condition implies linear growth, from (2.2), we have \(|F(x)|\le C_F(1+|x|)\). Here we keep the same notation \(C_F\) for simplicity. In the following, we also use \(|F(x)|\le C_F\sqrt{1+|x|^2}\). A similar bound holds for \(\sigma (x)\).
Hottovy et al. [14] assumed that the solutions are tight, that they just needed some continuity property of F and \(\sigma \), to pass the limit \(\epsilon \rightarrow 0\). Here we pose the Lipschitz assumption on F and \(\sigma \) to show the tightness of the solutions.
Next we present our main result.
Theorem 2.1
Under the assumptions of \((\mathbf {H_1})\)–\((\mathbf {H_3})\), for every \(t>0\) , \(\rho _t^\epsilon \), the solution to equation (1.8), converges weakly to \(\rho _t\) solving the following equation in weak sense
which corresponds to the following stochastic differential equation (SDE)
Here
and J(x) is the solution of the Lyapunov equation
Furthermore, there is a constant \(C_T>0\) such that for every \(t\in (0,T)\) and \(\psi \in C_0^\infty ({\mathbb {R}}^d)\)
where \(\Vert \cdot \Vert _{Lip}\) denotes the Lipschitz norm defined by
Remark 2.2
The above convergence rate in (2.7), from the estimate (4.11) in the proof, is sharp. So an interesting problem is the higher order correction to \(\rho _{t}^{\epsilon }\), that is what is the limit of
as \(\epsilon \rightarrow 0\) . To determine the limit we have to give a more detail estimation than that in Lemma 4.1. This will be considered in our future work.
Remark 2.3
The unique solution to equation (2.6) has the following explicit expression [2, Page 179]
In fact the matrix J(x) is the limit, as \(\epsilon \rightarrow 0\) , of the covariance of \(\sqrt{\epsilon }v^\epsilon _t\) with freezing \(x^\epsilon =x\) (Lemma 3.3) .
Remark 2.4
To see the relationship between (2.4) and (2.5), by Einstein summation notation, write (2.4) as
From (2.6), we have
Denote the right hand side of (2.8) by A, and extract the (i, j) element of both sides,
By the symmetry of J(x), we get
which corresponds to SDE (2.5).
To prove Theorem 2.1, we first show the tightness of \(\{x_t^\epsilon \}\) in Sect. 3, then for all sequences of \(\{\rho ^\epsilon _{\cdot }\}\) there exits a subsequence \(\{\rho ^{\epsilon _k}_{\cdot }\}\) converges weakly to \(\{\rho _{\cdot }\}\) as \(\epsilon _k\rightarrow 0\). Then, for the convergent subsequence \(\rho ^{\epsilon _k}_{\cdot }\) , we apply the averaging approach (Sect. 4) to the slow-fast system (1.8) to derive the limit equation.
The following lemma is used to give an explicit representation for the covariance of \(\sqrt{\epsilon } v^\epsilon _t\) with freezing \(x^\epsilon =x\) in Lemma 3.3 .
Lemma 2.1
[1, Theorem 2] Let \(I\subseteq {\mathbb {R}}\) be an open interval with \(t_0\in I\), \(A\in {\mathbb {C}}^{n\times n}\), \(B\in {\mathbb {C}}^{m\times m}\), \(C\in {\mathcal {C}}(I, {\mathbb {C}}^{n\times m})\) and \(D\in {\mathbb {C}}^{m\times n}\). The Lyapunov differential equation
has the unique solution
The following lemma is important in the averaging approach in Sect. 4 .
Lemma 2.2
[15, p. 120] Let \({\textbf{A}}=(a_{ij})_{1\le i,j\le d}\) and \({\textbf{u}}=(u_i)_{1\le i\le d}\) be \(d\times d\) matrix and \(d\times 1\) vector respectively. Each element of \({\textbf{A}}\) and \({\textbf{u}}\) is a function of \({\textbf{x}}=(x_1,x_2,\ldots ,x_d)\), then
where \(\textrm{grad}~{\textbf{u}}=\left( \frac{\partial u_i}{\partial x_j}\right) _{1\le i,j\le d}\) and \((\nabla \cdot {\textbf{A}})_j=\nabla \cdot {\textbf{A}}_j=\sum _{i=1}^d\frac{\partial a_{ij}}{\partial x_i}\) where \({\textbf{A}}_j\) is the j-th column of \({\textbf{A}}\).
3 Tightness of \(\{x^\epsilon \}_\epsilon \)
To prove the tightness of \(\{x^\epsilon \}_\epsilon \) in space \(C(0, T; {{\mathbb {R}}}^d)\), we need to show the boundedness in \(C(0, T; {{\mathbb {R}}}^d)\) and the Hölder continuity of \(\{x^\epsilon \}_\epsilon \) .
Lemma 3.1
Under assumptions \((\mathbf {H_1})\) and \((\mathbf {H_2})\), for all \(T > 0\),
and for \(0\le t_1\) , \(t_2\le T \),
Proof
We intend to write an expression of \(x^\epsilon _t\) in a mild formulation. Due to the state dependent friction, this is of some difficulty. For this we first consider the linear part of the \(v^\epsilon \)-equation (1.4), that is the following equation
Then
Integrating from 0 to \(\tau \) yields
and
Integrating from 0 to t for equation (3.3) yields
Now define
then
and
Thus we have [12, Lemma 4.2 of Chpter IV],
Similarly,
and
Then we derive
By Hölder inequality and Fubini theorem
By Doob’s maximal inequality, Fubini theorem and Hölder inequality, we obtain
Combining (3.5), (3.6) with (3.7) yields
Then Gronwall inequality yields
Next, let \(0\le t_1 < t_2\le T \),
First, Hölder inequality yields
Further by Hölder inequality, Fubini theorem and integral median theorem we have
In the last step of (3.9), we have used the fact that \(f(x)=xe^{-ax},a>0,x\in (0,+\infty )\) is bounded. Similarly, we have
Now (3.8)–(3.10) yields (3.2). The proof is complete. \(\square \)
Now by Lemma 3.1 and the Garcia–Rademich–Rumsey theorem [11], we have the tightness of solutions.
Lemma 3.2
The process \(\{x^\epsilon \}_\epsilon \) is tight in space \(C(0,T,{\mathbb {R}}^d)\) for all \(T>0\) .
Remark 3.1
The above result is assumed by Hottovy et al. [14, Assumption 3].
Next we show the limit of the covariance of \(\sqrt{\epsilon }v_t^\epsilon \) as \(\epsilon \rightarrow 0\) with frozen \(x^\epsilon =x\) . For this we consider the following linear equation for \(x\in {{\mathbb {R}}}^d\),
Then we have
Lemma 3.3
Assume \((\mathbf {H_1})\) and \((\mathbf {H_2})\) hold, for \(x\in {{\mathbb {R}}}^d\)
where \(|C(x,t)|\le C(1+|x|^2)\) and J(x) solves (2.6).
Proof
First by the Itô’s formula,
and
Applying Lemme 2.1 to equation (3.11) and the Duhamel’s principle to equation (3.12) respectively, yields
and
Thus
and then \(|F(x)\otimes {\mathbb {E}}v_t^{\epsilon ,x}| \le C(1+|x|^2)\). Now let \(\tau =\frac{t-s}{\epsilon }\), we have
The proof is complete. \(\square \)
Remark 3.2
Here we point out that an important step is to estimate \(\epsilon {\mathbb {E}}|v_t^\epsilon |^2\) in the work of Hottovy et al. [14] . However, in our approach we need the estimate of \(\epsilon {\mathbb {E}}v^{\epsilon ,x}_t\otimes v^{\epsilon , x}_t\) instead with fixed x .
4 Averaging Approach
In this section we just consider a convergent subsequence \(\rho ^{\epsilon _k}_{\cdot }\) and for simplicity we still write it as \(\rho ^\epsilon _{\cdot }\) . Let \(\rho _{\cdot }\) be the limit of \(\rho ^\epsilon _{\cdot }\) . Next we determine the equation for \(\rho _{\cdot }\) by an averaging approach.
Averaging is effective to study the approximation for a slow-fast system [7, 13, 16]. Here we apply the Khasminskii’s scheme [16] to (1.8). For small \(\epsilon \), \(\rho ^\epsilon _t\) evolves slow, so we can consider the fast part \(Y^\epsilon \) by freezing the slow part \(\rho ^\epsilon _t\) to be some \(\rho \in {\mathcal {P}}({{\mathbb {R}}}^d)\) and fix \(t=\tau \) in \({\mathbb {E}}^x(v_t^\epsilon \otimes v_t^\epsilon )\) . For this we introduce \({\tilde{Y}}_t^{\epsilon ,\rho ,\tau }(x)\) the solution of the following equation
with \({\tilde{Y}}_0^{\epsilon ,\rho ,\tau }(x)=Y_0\). The following lemma shows that the fast part converges uniformly in \(\tau \) to some vector with frozen slow part as \(\epsilon \rightarrow 0\) .
Lemma 4.1
For every fixed \(t_{*}>0\) , under the assumptions \((\mathbf {H_1})\)–\((\mathbf {H_3})\), fix \(\rho _t^{\epsilon }=\rho \in {\mathcal {P}}({{\mathbb {R}}}^{d})\) with \(\int |x|^2\rho (x) dx\) and \(\Vert |Y_0|\Vert _{L^1}\) bounded, there is a constant \(C_T>0\) such that for \(\varphi \in C_0^\infty ({\mathbb {R}}^d, {{\mathbb {R}}}^{d})\), and \(t\ge t_{*}\),
where
Proof
Applying Duhamel’s principle to equation (4.1) yields
Multiplying both sides of the equation (4.2) by the test function \(\varphi \) yields
By Hölder inequality,
Next,
by \(\mathbf {(H_1)}\) and \(\mathbf {(H_2)}\),
At last, by Lemma 3.3,
By Gaussian property and the definition of J(x), \(|\nabla _x\cdot (\rho (x)J(x))|\le C(1+|x|)\rho (x)\), then
and
Thus
By (4.3), (4.4) and (4.5), the proof is complete. \(\square \)
Remark 4.1
As we have mentioned in the Introduction, one can derive the limit equation formally for \(\rho _{t}\) by replacing \(Y^{\epsilon }_{t}\) by \(Y^{*, \rho }\) in the first equation of (1.8) .
However the slow part \(\rho ^\epsilon _t\) does evolve, in order to approximate \(Y^\epsilon _t\) we follow the Khasminskii’s scheme. For this we restrict the system in a small time interval, for example \([t_k, t_{k+1}]\) and freeze the slow part to be \(\rho ^\epsilon _{t_k}\) . We show that (Lemma 4.2) \(Y^\epsilon _t\) is approximated well by \({\hat{Y}}^\epsilon _t\) with frozen \(\rho ^\epsilon _t=\rho ^\epsilon _{t_k}\) if the length of time interval \([t_k, t_{k+1}]\) is small. For this we divide the time interval [0, T] into small intervals of size \(\delta >0\), i.e. \(0=t_0<t_1<\ldots <t_{\lfloor T/\delta \rfloor }+1=T\), \(t_k=k\delta \), \(k=0,1,\ldots , \lfloor T/\delta \rfloor \). For \(t\in [t_k,t_{k+1}]\), we define the auxiliary process \(\{ {\hat{\rho }}_t^\epsilon (x),{\hat{Y}}_t^\epsilon (x)\}_{0\le t \le T}\) satisfying
Remark 4.2
One can see that \({\hat{Y}}^\epsilon _t={\tilde{Y}}_t^{\epsilon , \rho ^\epsilon _{t_k}, t_k}\) .
Lemma 4.2
Assume \((\mathbf {H_1})\)–\((\mathbf {H_3})\) hold, for \(T>0\) and \(\varphi \in C_0^\infty ({\mathbb {R}}^d, {{\mathbb {R}}}^{d})\),
Proof
Let \(Z_t^\epsilon (x)=Y_t^\epsilon (x)-{\hat{Y}}_t^\epsilon (x)\). For all \(t\in [t_k, t_{k+1}]\), we have
By Duhamel’s principle,
For \(\varphi \in C_0^\infty ({\mathbb {R}}^d, {{\mathbb {R}}}^{d})\), we obtain
First
Then, by Lemma 3.1, we have
Further by Lemma 2.2,
Let \(g(x)=e^{-\frac{1}{\epsilon }\alpha ^\top (x)(t-s)}\varphi (x)\), by the chain rule,
Then, by assumptions \((\mathbf {H_1})\) and \((\mathbf {H_3})\),
thus
Similarly,
Then we have
The proof is complete. \(\square \)
Proof of Theorem 2.1
From the first equation of (1.8), for \(\psi \in C_0^\infty ({\mathbb {R}}^d)\) and \(\varphi =\nabla \psi \) we derive
First, by the expression of \(Y^{*,\rho }\),
Next, by Lemma 4.2,
by choosing \(\delta =O(\epsilon ^3)\).
Note that on the time interval \([t_k,t_{k+1}]\), \(\{{\hat{Y}}_t^\epsilon \}=\{{\tilde{Y}}_t^{\epsilon ,\rho _{t_k}^\epsilon ,t_k}\}\), let \(t_*\in [t_{i_0},t_{i_0+1}]\),
By Lemma 4.1,
By the defination of \(Y^{*,\rho }\) and Lemma 3.1,
By (4.9)–(4.12), passing the limit \(\epsilon \rightarrow 0\) yields
Since \(\varphi =\nabla _x \psi \), it yields
which is the weak form of
The proof is complete. \(\square \)
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Wang, M., Su, D. & Wang, W. Averaging on Macroscopic Scales with Application to Smoluchowski–Kramers Approximation. J Stat Phys 191, 22 (2024). https://doi.org/10.1007/s10955-024-03239-2
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DOI: https://doi.org/10.1007/s10955-024-03239-2