April 2023 Propagation of minimality in the supercooled Stefan problem
Christa Cuchiero, Stefan Rigger, Sara Svaluto-Ferro
Author Affiliations +
Ann. Appl. Probab. 33(2): 1588-1618 (April 2023). DOI: 10.1214/22-AAP1850


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 M1-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.

Funding Statement

The authors gratefully acknowledge financial support by the Vienna Science and Technology Fund (WWTF) under grant MA16-021 and by the FWF START-program under grant Y 1235.


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Christa Cuchiero. Stefan Rigger. Sara Svaluto-Ferro. "Propagation of minimality in the supercooled Stefan problem." Ann. Appl. Probab. 33 (2) 1588 - 1618, April 2023. https://doi.org/10.1214/22-AAP1850


Received: 1 October 2020; Revised: 1 March 2022; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692298
MathSciNet: MR4564435
Digital Object Identifier: 10.1214/22-AAP1850

Primary: 35Q84 , 60H30 , 60K35

Keywords: McKean–Vlasov equations , propagation of chaos , singular interactions , supercooled Stefan problem , systemic risk

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 2 • April 2023
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