Approximation of skew Brownian motion by snapping-out Brownian motions

TL;DR

This paper proves that snapping-out Brownian motions converge to skew Brownian motion as their permeability coefficients increase infinitely while maintaining a constant ratio, with convergence shown for semigroups, cosine families, and projections.

Contribution

It establishes a rigorous convergence result connecting snapping-out Brownian motions to skew Brownian motion under specific parameter limits.

Findings

Convergence of snapping-out Brownian motions to skew Brownian motion as permeability coefficients grow.

Demonstrates convergence of related semigroups, cosine families, and projections.

Provides a mathematical framework for approximating skew Brownian motion.

Abstract

We elaborate on the theorem saying that as permeability coefficients of snapping-out Brownian motions tend to infinity in such a way that their ratio remains constant, these processes converge to a skew Brownian motion. In particular, convergence of the related semigroups, cosine families and projections is discussed.