论文标题
超冷的Stefan问题中最小的传播
Propagation of minimality in the supercooled Stefan problem
论文作者
论文摘要
超冷的Stefan问题描述了物质的固体和液体相之间边界的演变，其中假定液体在其冰点以下冷却。遵循Delarue，Nadtochiy和Shkolnikov的方法，我们通过某个McKean-Vlasov方程构建了一阶段一维超冷的Stefan问题的解决方案，即使在爆炸的情况下，它也可以定义全球解决方案。 McKean-Vlasov方程的解决方案是作为通过打击时间相互作用的粒子系统的平均场限制而出现的，这对于系统性风险建模很重要。我们的主要贡献是：（i）我们证明了Skorokhod M1-Topology的一般紧密度定理，该定理适用于可以分解为连续和单调部分的过程。 （ii）我们证明，对于一般初始条件的粒子系统的扰动版本，我们证明了混乱的传播。 （iii）我们证明了Delarue，Nadtochiy和Shkolnikov的猜想，将所谓最小和物理解决方案的解决方案概念介绍，表明McKean-Vlasov方程的最小解决方案是物理的，只要初始条件是一种集成的。
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.
