On countably skewed Brownian motion with accumulation point

TL;DR

This paper establishes strong existence, uniqueness, and detailed properties of a special class of skewed Brownian motions with an accumulation point, connecting Dirichlet forms and stochastic calculus.

Contribution

It introduces a new class of countably skewed Brownian motions with an accumulation point, providing conditions for their key stochastic properties and applications.

Findings

Proved strong existence and pathwise uniqueness.

Derived conditions for non-explosion and recurrence.

Analyzed semimartingale properties and applications.

Abstract

In this work we connect the theory of Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.