PROPAGATION OF MINIMALITY IN THE SUPERCOOLED STEFAN PROBLEM

• 成果类型: Article
• 署名作者: Cuchiero, Christa; Rigger, Stefan; Svaluto-ferro, Sara
• 署名单位: University of Vienna; University of Verona
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1850
• 发表日期: 2023
• 页码: 1388-1418
• 关键词: singular interaction; systems

摘要

Supercooled Stefan problems describe the evolution of the boundary be-tween the solid and liquid phases of a substance, where the liquid is assumed to be cooled below its freezing point. Following the methodology of De-larue, 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 tight-ness theorem for the Skorokhod M1-topology which applies to processes that can be decomposed into a continuous and a monotone part. (ii) A propaga-tion 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.