- Le Chen
- Auburn University
- Date: Aug. 23, Tuesday, 2022
- Time: 2:30pm -- 3:30pm
- Room: Parker 326
- Abstract: The following one-dimensional stochastic wave equation has been studied as early as in the Walsh's notes 1980's:
\begin{align*} \partial_t^2 u(t,x) = \Delta u(t,x) + \lambda u(t,x) \dot{W}(t,x) \end{align*} where \(\dot{W}(t,x)\) refers to the space-time white noise and \(\lambda\ne 0\) is a constant. The upper bound for the moment Lyapunov exponents are known in the literature: <center> \( \displaystyle \limsup_{t\to\infty} t^{-1} \log \mathbb{E} \left[ u(t,x)^p \right] \le C p^{3/2},\quad p\ge 2.\) </center> In this talk, I will present a recent joint-work with Yuhui Guo and Jian Song arXiv:2206.10069 where we proved the corresponding lower bounds: <center> \(\displaystyle \liminf_{t\to\infty} t^{-1} \log \mathbb{E} \left[ u(t,x)^p \right] \ge C p^{3/2},\quad p\ge 2.\) </center>