Propagation of minimality in the supercooled Stefan problem

TL;DR

This paper develops a mathematical framework for analyzing supercooled Stefan problems using McKean-Vlasov equations, establishing global solutions and propagation of chaos, and proving a conjecture relating minimal and physical solutions.

Contribution

It introduces a new approach to construct global solutions for supercooled Stefan problems via McKean-Vlasov equations and proves a conjecture connecting minimal and physical solutions.

Findings

Proved a tightness theorem for the Skorokhod M1-topology.

Established propagation of chaos for a particle system.

Confirmed the conjecture relating minimal and physical solutions.

Abstract

Supercooled Stefan problems describe the evolution of the boundary between the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of Delarue, Nadtochiy and Shkolnikov, we construct solutions to the one-phase one-dimensional supercooled Stefan problem through a certain McKean-Vlasov equation, which allows to define global solutions even in the presence of blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of particle systems interacting through hitting times, which is important for systemic risk modeling. Our main contributions are: (i) we prove a general tightness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) We prove propagation of chaos for a perturbed version of the particle system for general initial conditions. (iii) We prove a conjecture of Delarue, Nadtochiy and Shkolnikov, relating the solution concepts of so-called minimal and physical solutions, showing that minimal solutions of the McKean-Vlasov equation are physical whenever the initial condition is integrable.