Date: Wednesday, January 19, 2022
Location: Zoom Virtual (4:00 PM to 5:00 PM)
Title: McKeanVlasov equations involving hitting times: blowups and global solvability
Abstract: We study two McKeanVlasov 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 FokkerPlanck equation has no blowup, and thus the McKeanVlasov dynamics is welldefined for all time $t \ge 0$. We take the PDE approach and develop a new comparison principle.
The second equation is $X(t) = X(0) + \beta t + B(t) + \alpha \log \mathbb{P}(\tau \le t)$, $t \ge 0$, whose FokkerPlanck equation is nonlocal. We prove that if $\beta,1/\alpha > 0$ are sufficiently large, the McKeanVlasov dynamics is welldefined for all time $t \ge 0$. The argument is based on a relative entropy analysis.
This is joint work with Erhan Bayraktar, Gaoyue Guo and Wenpin Tang.
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Speaker: Paul Zhang
Institution: UCSD
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