A random string with reflection in a convex domain

TL;DR

This paper investigates the behavior of a stochastic heat equation representing a random string confined within a convex domain, focusing on the reflection measure's structure using advanced probabilistic techniques.

Contribution

It introduces a novel analysis of the reflection measure for a stochastic heat equation in convex domains, employing infinite-dimensional integration by parts and weak convergence methods.

Findings

Explicit characterization of the reflection measure's Revuz measure

Application of infinite-dimensional integration by parts formula

Utilization of weak convergence results for log-concave invariant measures

Abstract

We study the motion of a random string in a convex domain $O$ in $\R^d$, namely the solution of a vector-valued stochastic heat equation, confined in the closure of $O$ and reflected at the boundary of $O$. We study the structure of the reflection measure by computing its Revuz measure in terms of an infinite-dimensional integration by parts formula. Our method exploits recent results on weak convergence of Markov processes with log-concave invariant measures.