Probabilistic approach to the strong Feller property
The authors introduce a new method for establishing the strong Feller property of diffusion processes in finite-dimensional and infinite-dimensional spaces. The main idea is to transfer the strong Feller property from a simpler, e.g. linear, equation by means of the Girsanov theorem. To this end, a characterization of strongly Feller Markov kernels in terms of equicontinuity of associated measures is given first. Then the main result is presented in the context of stochastic evolution equations on Hilbert spaces. The main two additional conditions for the transfer of the strong Feller property are that the considered process is already a Feller process and that the Radon-Nikodym derivative satisfies a certain uniform integrability condition. Numerous detailed examples, ranging from second order stochastic differential equations over stochastic delay equations to stochastic partial differential equations, show the power of the proposed method and give new results or provide shorter proofs than those obtained from analytical methods.
scientific article; zbMATH DE number 1546870