For given quasi-continuous (q.c.) functions g, h with g <= h and diffusion process M determined by stochastic differential equations or symmetric Dirichlet forms, characterizations of the value functions (e) over tildeg(s, x) = sup(sigma) J((s, x))(sigma) and (w) over bar (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.