Invariant measures for dichotomous stochastic differential equations in Hilbert spaces
The authors are interested in the existence of invariant measures for semilinear stochastic evolution equations in a real separable Hilbert space \(H\) \[ dX(t)=\{AX(t)+F[X(t)]\} dt+\sum_{i=0}^\infty\Phi_i[X(t)] dW_i(t),\quad X(0)=X_0, \quad t\geqslant 0,\tag{1} \] where \(A\) is the generator of a \(C_0\)-semigroup \((S(t))_{t\geqslant 0}\) on \(H\), the maps \(F\) and \(\Phi_i\), \(i\in\mathbb N\), are Lipschitz continuous from \(H\) to \(H\), and \(\{W_i\}_{i=0}^\infty\) is a sequence of independent scalar Wiener processes. Mild solutions being functions from \([0,\infty)\) to the Bochner space \(L^2(\Omega; H)\) are considered, where \((\Omega,\mathcal F,\mathbb P)\) is the underlying probability space. Assuming that the nonlinearities \(F\) and \(\Phi_i\) are Lipschitz continuous, infinite-dimensional noise that is white in time and colored in space is considered. It is shown that the existence of a bounded solution of (1) implies existence of an invariant measure of (1) if the equation (1) has an exponential dichotomy in the sense that the semigroup \((S(t))_{t\geqslant 0}\) is hyperbolic and the Lipschitz constants of \(F\) and \(\Phi_i\) are not too large.
semilinear stochastic evolution equations
scientific article; zbMATH DE number 1777422
scientific article