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COSAM Events 2022 September DMS Applied and Computational Mathematics Seminar

DMS Applied and Computational Mathematics Seminar

Time: Sep 30, 2022 (02:00 PM)

Location: 328 Parker Hall

Details:

Speaker: Yuming Paul Zhang

Title: McKean-Vlasov equations involving hitting times: blow-ups and global solvability

Abstract: We study two McKean-Vlasov equations involving hitting times. The equations are used in financial networks, computational neuroscience and some fluid models. The first equation is

$X(t) = X(0) + B(t) - \alpha \mathbb{P}(\tau \le t)$ , where

$(B(t); \, t \ge 0)$ denotes the Brownian motion, and

$\tau:= \inf\{t \ge 0: X(t) \le 0\}$ is the hitting time to zero of the process

$X$ . 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

$t \ge 0$ . 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 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 \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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