This paper is concerned with the analysis of blow-ups for two McKean–Vlasov equations involving hitting times. Let be standard Brownian motion, and be the hitting time to zero of a given process X. The first equation is . We provide a simple condition on α and the distribution of such that the corresponding Fokker–Planck equation has no blow-up, and thus the McKean–Vlasov dynamics is well defined for all time . 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 , , whose Fokker–Planck equation is nonlocal. We prove that for 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 . The argument is based on a new transform, which removes the nonlocal term, followed by a relative entropy analysis.
Ann. Appl. Probab.
34(1B):
1600-1622
(February 2024).
DOI: 10.1214/23-AAP1999
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