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

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.(1.2) 2 2n−1 L (n)2010 Mathematics Subject Classification. 60J60, 60K35, 60K37(primary), and 60J55(secondary).

Inverting the Ray-Knight identity on the line

“…that is to say the signs are opposite to those in (1.5). The reinforced diffusion in [21] can be considered as a dual of the self-repelling diffusion in the present paper.…”

“…Next we explain a non-rigorous heuristic derivation of an explicit solution to (1.2). A similar heuristic appears in the introduction to [21].…”

“…However, it turns out that the equation (1.2) is in some sense exactly solvable, and in this paper we will give the explicit solution which involves a divergent bifurcating stochastic flow of diffeomorphisms of R introduced by Bass and Burdzy in [3]. Our construction here is similar to that of [21], where we introduced a reinforced diffusion constructed out of a different, convergent, Bass-Burdzy flow.…”

“…The equation (1.2) is also somewhat misleading, as we believe that a solution q X t would not be a semi-martingale, admitting an adapted decomposition into a Brownian motion plus a drift term with zero quadratic variation, but with an infinite total variation. See [18,21] for a discussion on this point. However, it turns out that the equation (1.2) is in some sense exactly solvable, and in this paper we will give the explicit solution which involves a divergent bifurcating stochastic flow of diffeomorphisms of R introduced by Bass and Burdzy in [3].…”

“…constructing a Brownian motion with some conditioning on its family of local times, yet it is different and the processes obtained are different. Then, in [21] we constructed a linearly reinforced diffusion on R out of the flow of solutions to…”