On the oblique derivative problem for diffusion processes and diffusion equations with Hölder continuous coefficients

Abstract:

On a ${C^2}$-domain in a Euclidean space, we consider the oblique derivative problem for a diffusion equation and assume the coefficients of the diffusion and boundary operators are Hölder continuous. We then prove the uniqueness of diffusion processes and fundamental solutions corresponding to the problem. For the purpose, obtaining a stochastic representation of some solutions to the problem plays a key role; in our situation, a difficulty arises from the absence of a fundamental solution with ${C^2}$-smoothness up to the boundary. It is overcome by showing some stability of a fundamental solution and a diffusion process, respectively, under approximation of the domain. In particular, the stability of the fundamental solution is verified through construction: it is done by applying the parametrix method twice to a parametrix with explicit expression.