A Remark on Capacitary Measures for Diffusion Processes

In this note we shall give a characterization of capacitary measures of semipolar sets for diffusion processes, which is closely related to Dellacherie, Feyel and Mokobodzki2. In this study, the process X = (Xt) is assumed to be a standard process on (E, E), where E is a locally compact separable Hausdorff space and E is the Borel algebra on E. For X we suppose the following: i) X satisfies the strong duality hypotheses relative to a σ-finite measure ξ(dx) = dx; ii) U(x, K) and Û(x, K) are bounded on E for each compact subset K of E; iii) the kernel density u(x, y) of U(x, dy) relative to ξ is continuous off the diagonal and x → u(x, y), y → u(x, y) are lower semicontinuous. The brief explanation of the notations above will be given in the subsequent section. But we assume that the reader is familiar with terminologies in Blumenthal and Getoor1. A process is called a diffusion process on E if it is a standard process on (E, E) with continuous paths. For a subset B of E, TB denotes the hitting time of B, that is, TB = inf(t>0, XtεB).