The Nonlocal Stefan Problem via a Martingale Transport

TL;DR

This paper investigates the nonlocal Stefan problem using stochastic optimization, constructing global solutions and linking it to particle systems and obstacle problems, with new convergence results.

Contribution

It introduces a probabilistic approach to the nonlocal Stefan problem, connecting it to particle systems and establishing new convergence properties.

Findings

Constructed global-time weak solutions using stochastic optimization.

Provided a probabilistic interpretation of enthalpy and temperature variables.

Established exponential convergence for the melting problem.

Abstract

We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct global-time weak solutions and give a probabilistic interpretation for the solutions. An important ingredient in our analysis is a probabilistic interpretation of the enthalpy and temperature variables in terms of a particle system. Our approach in particular establishes the connection between the parabolic obstacle problem and the Stefan Problem for the nonlocal diffusions. For the melting problem, we show that our solution coincides with those studied in the literature, and obtain a new exponential convergence result.