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

TL;DR

This paper investigates conditions under which certain McKean-Vlasov equations involving hitting times avoid blow-ups, ensuring global well-posedness by connecting to Stefan problems and employing novel transforms and entropy methods.

Contribution

It provides new criteria for global existence of solutions to McKean-Vlasov equations with hitting times, linking them to Stefan problems and introducing a transform to handle non-local terms.

Findings

No blow-up for small enough lpha and suitable initial distributions.

Global well-posedness established for large positive drift eta and small lpha.

Connection made between McKean-Vlasov equations and supercooled Stefan problem.

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.