MCKEAN-VLASOV EQUATIONS INVOLVING HITTING TIMES: BLOW-UPS AND GLOBAL SOLVABILITY

This paper is concerned with the analysis of blow-ups for two McKean-Vlasov equations involving hitting times. Let (B(t); t >= 0) be standard Brownian motion, and tau := inf{t >= 0 : X(t) <= 0} be the hitting time to zero of a given process X. The first equation is X(t) = X(0-) + B(t) - alpha P(tau <= t). We provide a simple condition on a 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 >= 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 lnP(tau > t), t >= 0, whose FokkerPlanck equation is nonlocal. 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 >= 0. The argument is based on a new transform, which removes the nonlocal term, followed by a relative entropy analysis.