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

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 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 a non-explosive diffusion and its STMCA in terms of a thinness quantifier of the subdivision. An extension to absorbed diffusions uses a boundary-adapted Green-kernel thinness that remains finite even at certain singular exit boundaries. Under the stated regularity and subdivision hypotheses, this gives convergence at any rate strictly less than $\frac{1}{4}\wedge \frac{1}{p}$ in the appropriate thinness quantifier. For suitably adapted subdivisions satisfying a quadratic thinness bound, we show that the rate is doubled: we achieve any rate strictly less than $\frac{1}{2}\wedge \frac{2}{p}$ in terms of the maximum cell size. Additionally, we provide explicit analytical formulas for transition probabilities and times, enabling practical implementation of the STMCA, as well as easily verifiable sufficient non-explosion conditions for the bounds to hold. Numerical experiments illustrate our results.