Intermittency and stochastic pseudo-differential equation with spatially inhomogeneous white noise

  • Junfeng LiuEmail author


In this paper, we study the intermittent property for the following nonlinear stochastic partial differential equation (SPDE in the sequel) in (1+1)-dimension
$$\begin{aligned} \left( \frac{\partial }{\partial t}+q(x,D_x)\right) u(t,x)= g(u(t,x))\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x),\quad t>0 \quad \mathrm{and} \quad x\in {\mathbb {R}}, \end{aligned}$$
with \(q(x,D_x)\) is a pseudo-differential operator which generates a stable-like process. The forcing noise denoted by \(\frac{\partial ^2 w_\rho }{\partial t\partial x}(t,x)\) is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet \(w_\rho (t,x)\), we prove that the solution is weakly full intermittent based on the moment estimates of the solution.


Stochastic pseudo-differential equation Spatially inhomogeneous white noise Intermittency 

Mathematics Subject Classification

60H07 60H15 60G35 



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Authors and Affiliations

  1. 1.Department of StatisticsNanjing Audit UniversityNanjingPeople’s Republic of China

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