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COSAM Events 2025 February DMS Analysis and Stochastic Analysis Seminar (SASA)

DMS Analysis and Stochastic Analysis Seminar (SASA)

Time: Feb 26, 2025 (12:00 PM)

Location: 328 Parker Hall

Details:

Speaker: Wenxuan Tao (PhD student, University of Birmingham, UK)

Title: Stochastic heat equation in the non-Lipschitz regime

Abstract: Consider the stochastic heat equation (SHE)

$\frac{\partial u}{\partial t} = \frac{1}{2} \frac{\partial^2 u}{\partial x^2} + b(u) + \sigma(u)\dot{W}$ on the torus

$\mathbb{T} := [0,1]$ , which is driven by space-time white noise

$\dot{W}$ , subject to some nonnegative and nonvanishing initial condition

$u_0$ . It is known that when both

$b$ and

$\sigma$ are globally Lipschitz, there exists a unique solution to (SHE) for all time. Moreover, if both

$\sigma(0) = 0$ and

$b(0) = 0$ , the solution stays strictly positive almost surely for all time. On the other hand, if

$\sigma(u) \equiv 1$ is viewed as the limiting case of

$\sigma(u) = u^\alpha$ with

$\alpha \to 0$ , the solution for fixed

$(t,x)$ is a Gaussian random variable, which can take both positive and negative values. In this paper, we identify sufficient conditions on both

$b$ and

$\sigma$ to ensure the existence of a unique global solution that remains strictly positive while relaxing the global Lipschitz assumption. Canonical examples of such

$b$ and

$\sigma$ include

$b(u) = u (\log u)^{A_1}$ and

$\sigma(u) = u (\log u)^{A_2}$ with

$A_1 \in (0,1)$ and

$A_2 \in (0, 1/4)$ .

This is joint work with Le Chen and Jingyu Huang.

Host: Le Chen

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