Exit and Occupation times for Brownian Motion on Graphs with General Drift and Diffusion Constant

TL;DR

This paper derives general formulas for the exit and occupation times of a Brownian particle with spatially varying drift and diffusion on a graph with absorbing vertices, including boundary conditions and distribution analysis.

Contribution

It introduces a comprehensive framework for analyzing Brownian motion on graphs with general drift and diffusion, including boundary conditions and explicit formulas.

Findings

Derived formulas for average occupation times.

Analyzed exit time distributions and splitting probabilities.

Provided examples illustrating the theoretical results.

Abstract

We consider a particle diffusing along the links of a general graph possessing some absorbing vertices. The particle, with a spatially-dependent diffusion constant D(x) is subjected to a drift U(x) that is defined in every point of each link. We establish the boundary conditions to be used at the vertices and we derive general expressions for the average time spent on a part of the graph before absorption and, also, for the Laplace transform of the joint law of the occupation times. Exit times distributions and splitting probabilities are also studied and several examples are discussed.