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 $(\mathit{B}(\mathit{t});\mathit{t}\ge 0)$ be standard Brownian motion, and $\mathit{\tau }:=inf\{\mathit{t}\ge 0:\mathit{X}(\mathit{t})\le 0\}$ be the hitting time to zero of a given process X. The first equation is $\mathit{X}(\mathit{t})=\mathit{X}(0-)\mathbf{+}\mathit{B}(\mathit{t})-\mathit{\alpha }\mathbb{P}(\mathit{\tau }\le \mathit{t})$. We provide a simple condition on α and the distribution of $\mathit{X}(0-)$ such that the corresponding Fokker–Planck equation has no blow-up, and thus the McKean–Vlasov dynamics is well defined for all time $\mathit{t}\ge 0$. 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 $\mathit{X}(\mathit{t})=\mathit{X}(0-)\mathbf{+}\mathit{\beta }\mathit{t}\mathbf{+}\mathit{B}(\mathit{t})\mathbf{+}\mathit{\alpha }ln\mathbb{P}(\mathit{\tau }>\mathit{t})$, $\mathit{t}\ge 0$, whose Fokker–Planck equation is nonlocal. We prove that for $\mathit{\beta }>0$ 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 $\mathit{t}\ge 0$. The argument is based on a new transform, which removes the nonlocal term, followed by a relative entropy analysis.