DMS Analysis and Stochastic Analysis Seminar (SASA)

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