The reversible measures of multi-dimensional Ginzburg-Landau type continuum model
A stochastic dynamics for continuum field \(S(x)\) on \(R^ d\) with interactions prescribed by Ginzburg-Landau type Hamiltonian is investigated. The main problems discussed in the work are to clarify the structure of the family of reversible measures (r.m.'s) of this dynamics, namely (1) the characterization, (2) the construction, (3) showing the uniqueness of r.m.'s. The dynamics is defined by the following stochastic partial differential equation \[ dS_ t(x)=-{\mathcal A}S_ t(x)dt/2- V'(x,S_ t(x))/2+dw_ t(x),\tag{1} \] where \(S_ t\) determines a random time evolution of real-valued continuum field on \(R^ d\), \(w_ t\) is a cylindrical Brownian motion on \(L^ 2(R^ d)\), \({\mathcal A}\) is a symmetric differential operator of order \(2m\) having the form \[ {\mathcal A}f(x)=\sum_{|\alpha|,|\beta|\leq m}(- 1)^{|\alpha|}D^ \alpha\{a_{\alpha,\beta}D^ \beta f\}(x) \] with coefficients \(a_{\alpha,\beta}=a_{\beta,\alpha}\in C^ \infty(R^ d)\), \(V=V(x,s) : R^ d\times R\to R\) is a potential such that \(V(x,\cdot)\in C^ 1(R)\) for \(x\in R^ d\) and their derivatives \(V'=\partial V/\partial s\) in \(s\) are bounded and Lipschitz continuous. The reversibility property of the measure is formulated on the class \({\mathcal D}\) of ``simple'' functionals \(\Psi(S)=\psi(\langle S,\varphi_ 1\rangle,\dots,\langle S,\varphi_ k\rangle)\) with \(k=1,2,\dots,\;\psi=\psi(\alpha_ 1,\dots,\alpha_ k)\in C^ 2(R^ d)\) and \(\varphi_ 1,\dots,\varphi_ k\in C_ 0(R^ d)\), where \(\langle S,\varphi\rangle=\int_{R^ d}S(x)\varphi(x)dx\). A probability measure \(\mu\) is called a reversible one of the (1), if it satisfies \[ E^ \mu[\Phi(S_ 0)\Psi(S_ t)]=E^ \mu[\Psi(S_ 0)\Phi(S_ t)] \] for every \(t>0\) and \(\Phi\), \(\Psi\in{\mathcal D}\), where the superscript \(\mu\) means that \(\mu\) is the initial distribution of the solution \(S_ t\). The characterization problem is solved by the following proposition: The class of r.m.'s coincides with the class of Gibbs measures associated with potential \(V\).
stochastic partial differential equation
scientific article; zbMATH DE number 59703