Fine mesh limit of the VRJP in dimension one and Bass-Burdzy flow

TL;DR

This paper introduces the Linearly Reinforced Motion (LRM), a continuous space limit of the VRJP in one dimension, constructed via Bass-Burdzy flow, revealing new representations and symmetries in reinforced processes.

Contribution

It presents the first continuous space limit of VRJP in one dimension, constructed from Bass-Burdzy flow, and links it to ERRW, with new symmetry properties in initial local times.

Findings

LRM is a continuous limit of VRJP in 1D.

LRM can be represented as a mixture of diffusions.

The continuous limit can be derived from ERRW.

Abstract

We introduce a continuous space limit of the Vertex Reinforced Jump Process (VRJP) in dimension one, which we call Linearly Reinforced Motion (LRM) on $\R$. It is constructed out of a convergent Bass-Burdzy flow. The proof goes through the representation of the VRJP as a mixture of Markov jump processes. As a by-product this gives a representation in terms of a mixture of diffusions of the LRM and of the Bass-Burdzy flow itself. We also show that our continuous space limit can be obtained out of the Edge Reinforced Random Walk (ERRW), since the ERRW and the VRJP are known to be closely related. Compared to the discrete space processes, the LRM has an additional symmetry in the initial local times (initial occupation profile): changing them amounts to a deterministic change of the space and time scales.