Gibbsian dynamics and invariant measures for stochastic dissipative PDEs
type: Item
instance of: scholarly article
title: Gibbsian dynamics and invariant measures for stochastic dissipative PDEs
zbMATH Open document ID: 1023.60057
DOI: 10.1023/A:1019747716056
author: Di Liu
author: Weinan E
published in: Journal of Statistical Physics
publication date: 2003-08-06T00:00:00Z
review text: Let \(H\) be a separable Hilbert space and \(A: D(A)\to H\) a self-adjoint linear operator in \(H\) such that there exists an orthonormal basis \(\{e_{k}\}\) consisting of eigenvectors of \(A\), \(Ae_{k} = \lambda_{k}e_{k}\), \(0\leq \lambda_1\leq\lambda_2\leq\cdots\nearrow \infty\). Let \(W\) be a Wiener process in \(H\) of the form \(W(t) = \sum_{k\leq N} \sigma_{k}\beta_{k}(t)e_{k}\), where \(\{\beta _{k}\}\) are independent standard Wiener processes and \(\sigma_{k} \in \mathbb R\setminus\{0\}\). Let \(R: D(R)\to H\) be a nonlinear mapping. A stochastic differential equation \[ du = \{-Au + R(u)\} dt + dW \tag{1} \] in \(H\) is considered. It is supposed that there is a unique solution to (1) for any deterministic initial condition \(u(0) = u_0\in H\) and that the associated Markov process has a finite invariant measure. The authors aim at providing new sufficient conditions for uniqueness of this invariant measure. Assume that \(-\langle Ax,x\rangle + \langle R(x),x\rangle \leq -\eta|x|^2 + k_0\) for some constants \(\eta>0\), \(k_0\geq 0\) and all \(x\in D(A)\). Let there exist \(\alpha\in\mathopen[0,1\mathclose[\) and a nonnegative function \(K\) on \(H\) such that \[ \langle R(u_1)-R(u_2),u_1 -u_2\rangle \leq \alpha\langle Au_1-Au_2,u_1-u_2\rangle +K(u_1)|u_1-u_2|^2 \] for all \(u_1,u_2\in D(A)\), and \(\int_{H} K(u) d\mu(u)\leq\beta\) for some constant \(\beta\) and any invariant probability measure \(\mu\). Denote by \(P_\ell\) the orthogonal projection on the linear span of \(\{e_{k}\), \(k\leq N\}\). If \(N\) is sufficiently large so that \(-(1-\alpha)\lambda_{N}+\beta<0\) and two additional hypotheses on the function \(P_\ell R\) are satisfied, then the invariant measure for (1) is unique. These additional hypotheses might look rather complicated, but it is shown that the result is applicable to three one-dimensional stochastic partial differential equations: the stochastic Ginzburg-Landau equation, the stochastic Kuramoto-Sivashinsky equation, and the …
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 37L40
Mathematics Subject Classification ID: 37L55
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 76D99
Mathematics Subject Classification ID: 82C31
zbMATH DE Number: 1960204
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: invariant measures
zbMATH Keywords: ergodicity
MaRDI profile type: Publication
full work available at URL: a:1019747716056
OpenAlex ID: W1591993508
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description: scientific article; zbMATH DE number 1960204