Propagation of minimality in the supercooled Stefan problem

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) A general tightness theorem for the Skorokhod ${\mathit{M}}_1$-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) A propagation of chaos result for a perturbed version of the particle system for general initial conditions. (iii) The proof of 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.