On the gradient dynamics associated with wetting models

TL;DR

This paper studies the convergence of critical wetting models to Brownian motion variants, establishing continuous case results and proposing a conjecture on the limiting stochastic process involving Bessel SPDEs.

Contribution

It proves a continuous convergence result for wetting models and introduces a conjecture on the limiting process described by a Bessel SPDE.

Findings

Convergence of discrete wetting models to Brownian motion laws.

Approximation of reflecting Brownian motion by tilted Brownian meander.

A conjecture that the limiting process satisfies a Bessel SPDE.

Abstract

We consider several critical wetting models. In the discrete case, these probability laws are known to converge, after an appropriate rescaling, to the law of a reflecting Brownian motion, or of the modulus of a Brownian bridge, according to the boundary conditions. In the continuous case, a corresponding convergence result is proven in this paper, which allows to approximate the law of a reflecting Brownian motion by the law of Brownian meander tilted by its local time near the origin. On the other hand, these laws can be seen as the reversible probability measure of some Markov processes, namely, the dynamics which are encoded by integration by parts formulae. After proving the tightness of the associated reversible dynamics in the discrete case, based on heuristic considerations on the integration by parts formulae, we provide a conjecture on the limiting process, which we believe to satisfy a Bessel SPDE as introduced in a recent work by Elad Altman and Zambotti.