we study two mckean-vlasov equations involving hitting times. let $(b(t); t \geq 0)$ be standard brownian motion, and $\tau:= \inf\{t \geq 0: x(t) \leq 0\}$ be the hitting time to zero of a given process $x$. the first equation is $x(t) = x(0) + b(t) - \alpha \mathbb{p}(\tau \leq t)$.

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 $t \geq 0$. we take the pde approach and develop a new comparison principle.

the second equation is $x(t) = x(0) + \beta t + b(t) + \alpha \log \mathbb{p}(\tau \leq t)$, $t \geq 0$, whose fokker-planck equation is non-local. we prove that if $\beta,1/\alpha > 0$ are sufficiently large, the mckean-vlasov dynamics is well-defined for all time $t \geq 0$. the argument is based on a relative entropy analysis. this is joint work with erhan bayraktar, gaoyue guo and wenpin tang.