On an Optimal Stopping Problem of Time Inhomogeneous Diffusion Processes

For given quasi-continuous (q.c.) functions $g$, $h$ with $g\leq h$ and diffusion process {\bf M} determined by stochastic differential equations or symmetric Dirichlet forms, characterizations of the value functions $\tilde{e}_g(s,x) =\sup_\sigma J_{(s,x)}(\sigma)$ and $\bar{w}(s,x)=\inf_\tau\sup_\sigma J_{(s,x)}(\sigma,\tau)$ have been well studied. In this paper, by using the time-dependent Dirichlet forms, we generalize these results to time inhomogeneous diffusion processes. The difficulty of our case arises from the existence of essential semipolar sets [Y. Oshima, Tohoku Math. J. (2), 54 (2002), pp. 443-449]. In particular, excessive functions are not necessarily continuous along the sample paths. We get the result by showing such a continuity of the value functions.