Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces
The questions of existence and uniqueness of solutions to the semilinear stochastic differential equation \[ dX(t)= [AX+F(X)]dt+ dW(t),\qquad x\in H,\tag{1} \] in a Hilbert space \(H\) and their asymptotic behavior has been taken up time and again by different authors within different setups during the last three decades, cf. \textit{G. Da Prato} and \textit{J. Zabczyk} [``Stochastic equations in infinite dimensions'' (1992; Zbl 0761.60052)]. Distinguishing features of (1) are the size of the incremental covariance of the Wiener process \(W\) (nuclear or bounded), size, regularity, and domain of definition of the nonlinearity \(F\) (bounded, dissipative, Lipschitzian, continuous, measurable, defined everywhere or on a subspace) and quality of the linear operator \(A\) (selfadjoint or generator of a semigroup). In the present paper, \(A\) generates a \(C_0\)-semigroup \(S(t)\) on \(H\), \(F\) is bounded, defined everywhere on \(H\), and weakly continuous there. The incremental covariance \(Q\) is general. Let \(Q_t=\int^t_0 S(u)QS^*(u)du\) and let \(\mu\) be the invariant distribution on \(H\) of the linear equation associated with (1) (i.e., \(F=0\)). Assuming the conditions (A1) \(\sup_t\text{trace }Q_t<\infty\), (A2) \(\ker Q_t=\{0\}\), \(\text{range }S(t)\subseteq\text{range }\sqrt{Q_t}\), and (A3) \(\int^t_0|Q^{-1/2}_u S(u)|du<\infty\), the authors prove existence of a semigroup \((P_t)\) associated with (1) on \(L^p(\mu)\) and its related Sobolev space \(W^{1,p}(\mu)\). This semigroup is shown to have smoothing properties. They prove next the existence of a unique (mild) martingale solution \(X^x(t)\) to (1) in the sense of Da Prato, Zabczyk (loc. cit.) and the relationship \(P_t\varphi(x)=E\varepsilon(X^x(t))\). They finally show that \(X\) possesses a unique invariant measure and its absolute continuity with respect to \(\mu\) with densities in all spaces \(L^p(\mu)\), \(p<\infty\). The methods of proof are perturbation theory of semigroups, irreducibility, and the strong Feller property.
semilinear stochastic differential equations
existence/uniqueness of invariant measures
scientific article; zbMATH DE number 784421
scientific article