McKean-Vlasov equations involving hitting times: blow-ups and global solvability

Abstract

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 $\tau :=\inf \{t\ge 0:X(t)\le 0\}$ be the hitting time to zero of a given process $X$. The first equation is $X(t)=X(0)+B(t)-\alpha \mathbb{P}(\tau \le t)$. We provide a simple condition on $\alpha$ 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 $X(t)=X(0)+\beta t+B(t)+\alpha \log \mathbb{P}(\tau >t)$, whose Fokker-Planck equation is non-local. We prove that for $\beta >0$ sufficiently large and $\alpha$ 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.

22 pages

References in corpus (1)

• Global solutions to the supercooled Stefan problem with blow-ups: regularity and uniqueness

Cited by in corpus (1)

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