A Gibbs variational principle in space-time for infinite-dimensional diffusions
Consider on \(\Omega:={\mathcal C}(\mathbb{R},\mathbb{R})^{\mathbb{Z}^d}\) the infinite system of stochastic differential equations: for \(\gamma\in\mathbb{Z}^d\) and \(t\in(0,1]\) \[ dX^j_t=dB^j_t+ [\varphi'(X^j_t)+ F(X\circ \Theta_t^j)]dt,\tag{*} \] where the \(B^j\)'s are independent Brownian motions, \(\varphi\) is such that if \(F\equiv 0\), then the \(X^j\)'s are (i.i.d.) real ergodic processes, \(\Theta^j_t\) is the space-time translation on \(\Omega\), and \(F\) is real measurable bounded on \({\mathcal C}(\mathbb{R}-,\mathbb{R})^{\mathbb{Z}^d}\) and depends only on a finite number of coordinates. Denote by \(\mathbb{P}\) the stationary law on \(\Omega\) solving (*) in the case \(F\equiv 0\), and consider some space-time invariant law \(Q\) on \(\Omega\), which has finite entropy with respect to \(\mathbb{P}\). Then it is proved that \(Q\) is a weak solution to (*) if and only if it is a Gibbs state with respect to the Hamiltonian \(\sum_j[\int F\circ \Theta^j_t\cdot d\widetilde\beta^j_t-\tfrac 12\int F^2\circ \Theta^j_t\cdot dt]\), and also if and only if it minimizes some free energy functional associated with \(\mathbb{P}\) and \(F\).
scientific article; zbMATH DE number 1761131