Diffusion spiders: Green kernel, excessive functions and optimal stopping

TL;DR

This paper studies diffusion spiders, a type of Markov process on graphs, deriving explicit formulas for their resolvent kernels and excessive functions, and applying these to solve optimal stopping problems.

Contribution

It provides explicit formulas for the resolvent kernel and excessive functions of diffusion spiders, advancing the understanding of their potential theory and optimal stopping solutions.

Findings

Explicit density of the resolvent kernel derived.

Representation measure for excessive functions obtained.

Optimal stopping problems solved explicitly.

Abstract

A diffusion spider is a strong Markov process with continuous paths taking values on a graph with one vertex and a finite number of edges (of infinite length). An example is Walsh's Brownian spider where the process on each edge behaves as Brownian motion. We calculate the density of the resolvent kernel in terms of the characteristics of the underlying diffusion. Excessive functions are studied via the Martin boundary theory. The main result is an explicit expression for the representing measure of a given excessive function. These results are used to solve optimal stopping problems for diffusion spiders.