A uniqueness theorem for the martingale problem describing a diffusion in media with membranes

TL;DR

This paper establishes a uniqueness theorem for a martingale problem modeling a diffusion process in a space with a membrane that skews and delays the process, ensuring well-posedness of the model.

Contribution

It introduces a martingale problem for diffusions with membranes and proves the uniqueness of its solution, advancing the mathematical understanding of such processes.

Findings

Proved the existence of at most one solution to the martingale problem.

Formulated a model for diffusion with membranes on smooth surfaces.

Ensured the well-posedness of diffusion models with skewing and delaying membranes.

Abstract

We formulate a martingale problem that describes a diffusion process in a multidimensional Euclidean space with a membrane located on a given smooth surface and having the properties of skewing and delaying. The theorem on the existence of no more than one solution to the problem is proved.