- Wenxuan Tao (Ph.D. student)
- University of Birmingham, UK
- Date: Feb. 26, Wednesday, 2025
- Time: 12:00pm -- 12:50pm
- Host: Le Chen
- Room: Parker 328
- Abstract: Consider the stochastic heat equation (SHE) \(\displaystyle \frac{\partial u}{\partial t} = \frac{1}{2} \frac{\partial^2 u}{\partial x^2} + b(u) + \sigma(u)\dot{W}\) 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 exits 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 \(\sigma(u) \equiv 1\) 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 \(\sigma = u (\log u)^{A_2}\) with \(A_1 \in (0,1)\) and \(A_2 \in (0, 1/4)\). This is a joint work with Le Chen and Jingyu Huang.