Invariant measures for uniformly recurrent diffusion kernels (Q2266086)

The aim of the paper is to characterize invariant measures for a uniformly recurrent diffusion kernel T on a locally compact Hausdorff space X. The main theorem can be summarized in the following way: Denote by H(T) the cone generated by non-negative T-invariant measures and put \(X_ 0=cl(\cup_{\mu \in H(T)}\sup p \mu).\) Then there exists a strictly positive diffusion kernel W on \(X_ 0\), uniquely determined except for the equivalence of diffusion kernels, such that \(TW=W\) and H(T) coincides with W-potentials. Further the author investigates, for example, the case when H(T) is one dimensional, invariant measures for diffusion semi-groups, Hunt diffusion kernels and diffusion kernels of the convolution type.