Gibbsian dynamics and invariant measures for stochastic dissipative PDEs
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 stochastic Cahn-Hilliard equation.
stochastic partial differential equations
scientific article; zbMATH DE number 1960204