On Portenko's approximation of skew Brownian motion

TL;DR

This paper generalizes Portenko's approximation theorem for skew Brownian motion using operator semigroup theory, extending it to Walsh processes on star graphs and analyzing the convergence through Sturm--Liouville perturbations.

Contribution

It extends Portenko's classical theorem to Walsh processes on star graphs, providing a new perspective via operator semigroup convergence and singular Sturm--Liouville perturbations.

Findings

Proves convergence of Feller semigroups in the generalized setting

Establishes a simple transformation of Walsh process parameters

Links Portenko's approximation to Sturm--Liouville singular perturbations

Abstract

From the perspective of the theory of operator semigroups, we reflect back on the classical theorem of Portenko devoted to approximation of skew Brownian motion. The theorem says that by concentrating the power of drift of a diffusion process around a point one obtains an equivalent of a semi-permeable membrane at this point, described by skew Brownian motion's boundary condition. We prove convergence of the corresponding Feller semigroups and in doing so, generalize Portenko's theorem to the case of the Walsh processes on star graphs. Our analysis leads through singular perturbations of Sturm--Liouville equations, and reveals that as a result of Portenko-type approximation parameters of Walsh processes are transformed in a simple and elegant manner.