A martingale problem associated with diffusion operators in a domain
Norio Okada · Kodai Mathematical Journal · 1981
For a given diffusion operator with continuous coefficients, we consider the solution P StX of the martingale problem such that all paths starting from x at time s stay in a domain Z) in [0, oo)χR d for ever.That is, the support of P s , x is contained in C([0, oo) } D).Under some regularity conditions, we shall obtain a necessary and sufficient condition for the existence of such a solution on D. Results and precise formulations are stated in § 2. In § 3, the results of § 2 and some lemmas for them will be proved.In §4, we shall consider some examples such that the uniqueness of solutions of § 2 holds.In these examples sample paths stay on the boundary after hitting the boundary.To discuss processes with reflecting or entrance boundary, we need probably another approach.I am grateful to Professors M. NAGASAWA and M. Motoo for their kind comments and advice.§ 2. Existence and related topics.Let D be an open subset with the boundary 3D of [0, cχD)χR d and assume that:x)R d is bounded and continuous.