Denote by Xt an n -dimensional symmetric Markov process associated with an elliptic operator [FORMULA OMMITED] where (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f (x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied: (i) For almost every t ∈ [0, 1], f (., t) ∈ H10(Rn) and ∫01 ǀ∇x ǀ2 dx dt + ∫10 ∫ǀǀ2 dx dt < ∞. (ii) Let τm: 0 =tm0 < tm1 <.…< tmv(m) = 1 be a sequence of subdivisions of [0,1] so that [FORMULA OMMITED] Then [FORMULA OMMITED] As an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt associated with the elliptic operator [FORMULA OMMITED] where (bi,) are bounded measurable functions of x and we suppose that p(x, 0) ∈ C10(Rn). Then, p(Yt, t) is a regular Dirichlet process and therefore p(…) satisfies (i) and (ii). © 1990, Royal Society of Edinburgh. All rights reserved.
