31. chen.guo.ea:22:moments#

Moments and asymptotics for a class of SPDEs with space-time white noise

Le Chen, Yuhui Guo, and Jian Song

To appeare in Transactions of American Mathematical Scociety

Abstract: In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition:

\[\left(\partial^{\beta}_t+\dfrac{\nu}{2}\left(-\Delta\right)^{\alpha / 2}\right) u(t, x) = \: I_{t}^{\gamma}\left[\lambda u(t, x) \dot{W}(t, x)\right] \quad t>0,\: x\in\mathbb{R}^d,\]

with constants \(\lambda\ne 0\) and \(\nu>0\), where \(\partial^{\beta}_t\) is the Caputo fractional derivative of order \(\beta\in(0,2]\), \(I_{t}^{\gamma}\) refers to the Riemann-Liouville integral of order \(\gamma \ge 0\), and \(\left(-\Delta\right)^{\alpha/2}\) is the standard fractional/power of Laplacian with \(\alpha>0\). We concentrate on the scenario where the noise \(\dot{W}\) is the space-time white noise. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang’s condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the \(p\)-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the \(p\)-th moment Lyapunov exponents. In particular, by letting \(\beta = 2\), \(\alpha = 2\), \(\gamma = 0\), and \(d = 1\), we confirm the following standing conjecture for the stochastic wave equation:

\[\frac{1}{t}\log\mathbb{E}[|u(t,x)|^p ] \asymp p^{3/2}, \quad \text{for } p\ge 2 \text{ as } t\to \infty.\]

The method for the lower bounds is inspired by a recent work of , where the authors focus on the space-time colored Gaussian noise case.

MSC 2010 subject classifications: Primary 60H15; Secondary 60G60, 26A33, 37H15, 60H07.

Keywords: stochastic partial differential equation, stochastic heat/wave equation, space-time white noise, Dalang’s condition, moment asymptotics, intermittency, moment Lyapunov exponent.

Preprint

[CGS22] Le Chen, Yuhui Guo & Jian Song (2022) ‘Moments and asymptotics for a class of SPDEs with space-time white noise’, preprint arXiv:2206.10069, to appear in Trans. Amer. Math. Soc.

@article{chen.guo.ea:22:moments,
  title         = {Moments and asymptotics for a class of SPDEs with space-time white noise},
  author        = {Le Chen and Yuhui Guo and Jian Song},
  year          = {2022},
  month         = {June},
  journal       = {preprint arXiv:2206.10069, to appear in Trans. Amer. Math. Soc.},
  url           = {https://www.arxiv.org/abs/2206.10069}
}

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