Diffusion spiders: Green kernel, excessive functions and optimal stopping

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 a Brownian motion. In this paper we calculate firstly the density of the resolvent kernel in terms of the characteristics of the underlying diffusion. Excessive functions are studied via the Martin boundary theory. A crucial result is an expression for the representing measure of a given excessive function. These results are used to solve optimal stopping problems for diffusion spiders.(c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper investigates the characteristics and properties of diffusion spiders and calculates the density of the resolvent kernel. The study of excessive functions leads to the expression of the representing measure for a given excessive function. These results are then applied to solving optimal stopping problems for diffusion spiders.