论文标题
一类SPDE的矩和渐近学，带有时空白噪声
Moments and asymptotics for a class of SPDEs with space-time white noise
论文作者
论文摘要
在本文中，我们考虑了具有恒定初始条件的时空和时间变量中分数顺序的非线性随机部分微分方程： \ begin {equation*} \左（\ partial^β_t+\dfracν{2} \ left（-Δ\ right）^{α/ 2} \ right） \ quad t> 0，〜x \ in \ mathbb r^d， \ end {equation*} $ \ dot {w} $是时空白噪声，$α> 0 $，$β\ in（0,2] $，$γ\ ge 0 $，$λ\ neq0 $和$ν> 0 $。在Dalang的条件下获得了二次exports wore wore wore wore exports wo exports wore wore wo exports ye declast wo in. exports wore wore wo. skorhohodsence section wore wer。推导$ p $ them tomm tounds，并找到匹配的下限。 t^{ - 1} \ log \ mathbb e [u（t，x）^p] \ asymp p^{3/2}，\ quad \ text {for $ p \ ge 2 $ as $ t $ as $ t \ t \ t \ to \ infty $。} \ end {align*} 下限的方法的灵感来自Hu和Wang [HW21]的最新作品，在该工作中，作者专注于时空彩色高斯噪声。
In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \begin{equation*} \left(\partial^β_t+\dfracν{2}\left(-Δ\right)^{α/ 2}\right) u(t, x)= ~ I_{t}^γ\left[λu(t, x) \dot{W}(t, x)\right] \quad t>0,~ x\in\mathbb R^d, \end{equation*} where $\dot{W}$ is space-time white noise, $α>0$, $β\in(0,2]$, $γ\ge 0$, $λ\neq0$ and $ν>0$. 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 $β=2$, $α=2$, $γ=0$, and $d=1$, we confirm the following standing conjecture for the stochastic wave equation: \begin{align*} t^{-1}\log\mathbb E[u(t,x)^p] \asymp p^{3/2}, \quad \text{for $p\ge 2$ as $t\to \infty$.} \end{align*} The method for the lower bounds is inspired by a recent work by Hu and Wang [HW21], where the authors focus on the space-time colored Gaussian noise.
