A note on intermittency for the fractional heat equation

Abstract

The goal of the present note is to study intermittency properties for the solution to the fractional heat equation $$\frac{\partial u}{\partial t}(t,x) = -(-Δ)^{β/2} u(t,x) + u(t,x)\dot{W}(t,x), \quad t>0,x \in \bR^d$$ with initial condition bounded above and below, where $\beta \in (0,2]$ and the noise $W$ behaves in time like a fractional Brownian motion of index $H>1/2$, and has a spatial covariance given by the Riesz kernel of index $\alpha \in (0,d)$. As a by-product, we obtain that the necessary and sufficient condition for the existence of the solution is $\alpha <\beta$.

12 pages