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2020 Representations of the Vertex Reinforced Jump Process as a mixture of Markov processes on $\mathbb {Z}^{d}$ and infinite trees
Thomas Gerard
Electron. J. Probab. 25: 1-45 (2020). DOI: 10.1214/20-EJP510

Abstract

This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. In [21], it was shown that the VRJP on finite graphs, under a certain time rescaling, has the distribution of a mixture of Markov jump processes. This representation was extended to infinite graphs in [23], by introducing a random potential $\beta $. In this paper, we show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form as in [23], using the random field $\beta $ and harmonic functions for an associated operator $H_\beta $. This allows to show that the VRJP on $\mathbb {Z}^{d}$ (with certain initial conditions) has a unique representation, by proving that an associated Martin boundary is trivial. Moreover, on infinite trees, we construct a family of representations, that are all different when the VRJP is transient and the tree is $d$-regular (with $d\geq 3$).

Citation

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Thomas Gerard. "Representations of the Vertex Reinforced Jump Process as a mixture of Markov processes on $\mathbb {Z}^{d}$ and infinite trees." Electron. J. Probab. 25 1 - 45, 2020. https://doi.org/10.1214/20-EJP510

Information

Received: 23 July 2019; Accepted: 10 August 2020; Published: 2020
First available in Project Euclid: 12 September 2020

zbMATH: 07252702
MathSciNet: MR4150520
Digital Object Identifier: 10.1214/20-EJP510

Subjects:
Primary: 31C35, 60J75, 60K37

JOURNAL ARTICLE
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