论文标题
McKean-Vlasov方程式涉及打击时间：爆炸和全球解决性
McKean-Vlasov equations involving hitting times: blow-ups and global solvability
论文作者
论文摘要
本文涉及两个涉及打击时间的McKean-Vlasov方程的爆炸分析。令$（b（t）; \，t \ ge 0）$为标准的布朗尼运动，$τ：= \ inf \ {t \ ge 0：x（t）\ le 0 \} $是给定流程$ x $的零时间。第一个方程为$ x（t）= x（0） + b（t）-α\ mathbb {p}（τ\ le t）$。我们提供了$α$的简单条件，并且分布$ x（0）$，使得相应的fokker-planck方程没有爆炸，因此McKean-Vlasov Dynamics在所有时间$ t \ ge ge 0 $均已明确定义。我们的方法依赖于McKean-Vlasov方程与超冷的Stefan问题以及几种比较原则之间的联系。第二个方程为$ x（t）= x（0） +βT + b（t） +α\ log \ mathbb {p}（τ> t）$，其fokker-planck方程是非本地的。我们证明，对于$β> 0 $，足够大，$α$不超过足够小的正常常数，没有爆炸，McKean-Vlasov Dynamics一直定义为有史以来$ t \ ge 0 $。该参数基于一个新的转换，该转换删除了非本地术语，然后进行相对熵分析。
This paper is concerned with the analysis of blow-ups for two McKean-Vlasov equations involving hitting times. Let $(B(t); \, t \ge 0)$ be standard Brownian motion, and $τ:= \inf\{t \ge 0: X(t) \le 0\}$ be the hitting time to zero of a given process $X$. The first equation is $X(t) = X(0) + B(t) - α\mathbb{P}(τ\le t)$. We provide a simple condition on $α$ and the distribution of $X(0)$ such that the corresponding Fokker-Planck equation has no blow-up, and thus the McKean-Vlasov dynamics is well-defined for all time $t \ge 0$. Our approach relies on a connection between the McKean-Vlasov equation and the supercooled Stefan problem, as well as several comparison principles. The second equation is $X(t) = X(0) + βt + B(t) + α\log \mathbb{P}(τ> t)$, whose Fokker-Planck equation is non-local. We prove that for $β> 0$ sufficiently large and $α$ no greater than a sufficiently small positive constant, there is no blow-up and the McKean-Vlasov dynamics is well-defined for all time $t \ge 0$. The argument is based on a new transform, which removes the non-local term, followed by a relative entropy analysis.
