The Annals of Probability

Markov processes on time-like graphs

Krzysztof Burdzy and Soumik Pal

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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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Ann. Probab., Volume 39, Number 4 (2011), 1332-1364.

First available in Project Euclid: 5 August 2011

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Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60G60: Random fields 60J99: None of the above, but in this section

Harness graphical Markov model time-like graphs


Burdzy, Krzysztof; Pal, Soumik. Markov processes on time-like graphs. Ann. Probab. 39 (2011), no. 4, 1332--1364. doi:10.1214/10-AOP583.

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