间歇性
乘性噪声
数学
乘法函数
力矩(物理)
面积二阶矩
维数(图论)
数学分析
度量(数据仓库)
李雅普诺夫指数
订单(交换)
噪音(视频)
数学物理
组合数学
物理
量子力学
热力学
非线性系统
湍流
信号传递函数
人工智能
模拟信号
图像(数学)
计算机科学
数字信号处理
数据库
工程类
几何学
财务
电气工程
经济
作者
Quentin Berger,Carsten Chong,Hubert Lacoin
出处
期刊:Cornell University - arXiv
日期:2021-01-01
标识
DOI:10.48550/arxiv.2111.07988
摘要
We study the stochastic heat equation (SHE) $\partial_t u = \frac12 Δu + βu ξ$ driven by a multiplicative Lévy noise $ξ$ with positive jumps and amplitude $β>0$, in arbitrary dimension $d\geq 1$. We prove the existence of solutions under an optimal condition if $d=1,2$ and a close-to-optimal condition if $d\geq3$. Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of $u$, we further show that the solution has finite $p$th moments for $p>0$ whenever the noise does. Finally, for any $p>0$, we derive upper and lower bounds on the moment Lyapunov exponents of order $p$ of the solution, which are asymptotically sharp in the limit as $β\to0$. One of our most striking findings is that the solution to the SHE exhibits a property called strong intermittency (which implies moment intermittency of all orders $p>1$ and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity $β>0$, in any dimension $d\geq1$.
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