Markov processes on time-like graphs

TL;DR

This paper introduces Markov processes on directed graphs representing 'time', establishing conditions for their independence and analyzing special processes called harnesses, including their continuum limits.

Contribution

It develops a framework for Markov processes on time-like graphs, characterizes their independence properties, and studies harnesses and their limits.

Findings

Conditions for Markov processes on specific graphs like hexagonal lattices.

Identification of harnesses as a key class of processes on time-like graphs.

Analysis of continuum limits of harnesses.

Abstract

We study Markov processes where the "time" parameter is replaced by paths in a directed graph from an initial vertex to a terminal one. Along each directed path the process is Markov and has the same distribution as the one along any other directed path. If two directed paths do not interact, in a suitable sense, then the distributions of the processes on the two paths are conditionally independent, given their values at the common endpoint of the two paths. Conditions on graphs that support such processes (e.g., hexagonal lattice) are established. Next we analyze a particularly suitable family of Markov processes, called harnesses, which includes Brownian motion and other L\'{e}vy processes, on such time-like graphs. Finally we investigate continuum limits of harnesses on a sequence of time-like graphs that admits a limit in a suitable sense.