On sequentially weakly Feller solutions to SPDE's
A particular class of SPDE's defining weakly Feller transition semigroups was investigated by A. Ichikawa (1984). He considered equations whose nonlinear terms depend only on a projection of the solution to a finite-dimensional subspace and have finite-dimensional ranges. Markov Feller processes with respect to the weak topology were studied by \textit{G. Leha} and \textit{G. Ritter} [Stochastics Stochastics Rep. 48, No. 3/4, 195-225 (1994; Zbl 0828.60063)]. In this note the authors find a general class of semilinear stochastic evolution equations whose transition semigroups map bounded weakly continuous ones; let us call such semigroups sequentially weakly Feller. In Section 2 the authors prove that any Ornstein-Uhlenbeck process is sequentially weakly Feller and that the same assertion holds for the Markov process defined by a semilinear stochastic equation \[ dx=\bigl( Ax+f(x)\bigr)dx +\sigma(x) Q^{1/ 2}dw \] provided \(A\) generates a compact semigroup and the nonlinear terms \(f, \sigma\) are Lipschitz continuous. Also they discuss that the assumptions on the drift can be relaxed by means of the Girsanov theorem, the compactness of the semigroup generated by \(A\) again playing an important role. In Section 3 the sequential weak Feller property is shown to be sufficient for employing the Krylov-Bogolyubov procedure and some examples to which our results are applicable are provided.
stochastic partial differential equations
scientific article; zbMATH DE number 1822106