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July 2011 Markov processes on time-like graphs
Krzysztof Burdzy, Soumik Pal
Ann. Probab. 39(4): 1332-1364 (July 2011). DOI: 10.1214/10-AOP583

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 Lévy 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.

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Krzysztof Burdzy. Soumik Pal. "Markov processes on time-like graphs." Ann. Probab. 39 (4) 1332 - 1364, July 2011. https://doi.org/10.1214/10-AOP583

Information

Published: July 2011
First available in Project Euclid: 5 August 2011

zbMATH: 1234.60089
MathSciNet: MR2857242
Digital Object Identifier: 10.1214/10-AOP583

Subjects:
Primary: 60G60 , 60J60 , 60J99

Keywords: graphical Markov model , Harness , time-like graphs

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.39 • No. 4 • July 2011
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