Intermittence and time fractional stochastic partial differential equations

We consider time fractional stochastic heat type equation $\partial^β_tu(t,x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[σ(u)\stackrel{\cdot}{W}(t,x)]$ in $(d+1)$ dimensions, where $ν>0$, $β\in (0,1)$, $α\in (0,2]$, $d<\min\{2,β^{-1}\}\a$, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2}$ is the generator of an isotropic stable process, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $σ:\RR{R}\to\RR{R}$ is Lipschitz continuous. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove: (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Foondun and Khoshnevisan \cite{foondun-khoshnevisan-09} %(Mohammud Foondun and Davar Khoshnevisan, Intermittence and nonlinear parabolic %stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568) and Conus and Khoshnevisan \cite{conus-khoshnevisan} % (On the existence and position of the farthest peaks of a family of stochastic %heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, 681--701) on the parabolic stochastic heat equations.

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