General diffusions on metric graphs as limits of time-space Markov Chains

TL;DR

This paper introduces a new approximation method called STMCA for diffusions on metric graphs, providing convergence bounds, explicit formulas, and numerical validation for practical simulation of such processes.

Contribution

The paper develops the STMCA, a novel Markov chain approximation for diffusions on metric graphs, with explicit formulas and proven convergence bounds.

Findings

Convergence occurs at rates depending on subdivision fineness.

Explicit formulas enable practical implementation.

Numerical experiments validate theoretical results.

Abstract

We introduce the Space-Time Markov Chain Approximation (STMCA) for a general diffusion process on a finite metric graph $\Gamma$. The STMCA is a doubly asymmetric (in both time and space) random walk defined on a subdivisions of $\Gamma$, with transition probabilities and conditional transition times that match, in expectation, those of the target diffusion. We derive bounds on the $p$-Wasserstein distances between the diffusion and its STMCA in terms of a thinness quantifier of the subdivision. This bound shows that convergence occurs at any rate inferior to $\frac{1}{4} \wedge \frac{1}{p} $ in terms of the the maximum cell size of the subdivision, for adapted subdivisions, at any rate inferior to $\frac{1}{2} \wedge \frac{2}{p} $. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA. Numerical experiments illustrate our results.