The construction of a class of diffusions
Donald A. Dawson · Illinois Journal of Mathematics · 1964
Introduction E. B. Dynkin [4] has shown that the generator of a diffusion on a locally compact, separable space Q has a canonical representation in terms of the mean hitting times and hitting probabilities.Let x(t) be a strict Markov process with generator @ whose domain is D(@).Let f e D(@), ( e Q, U be a U neighborhood of with compact closure and nonnull boundary and r beIt is easy to show that @ satisfies a maximum property and is a local operator on C(Q).W. Feller [6] has posed the converse question, namely, does every local operator on C(Q) which satisfies the maximum property generate a dif- fusion.As a partial solution of this problem it will be shown that every such operator arising from a set of mean hitting times and hitting probabilities having certain smoothness properties does indeed generate a diffusion.The method employed is the construction of a sequence of approximating random walks which will be shown to converge to a limit process which is a diffusion.This is an extension of the construction of F. B. Knight [10], [11] for the one- dimensional case.This paper is based on he author's Ph.D.