Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Abstract

We consider the stochastic heat equation with multiplicative noise $u_t={1/2}Δu+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $α$, or the Bessel kernel of order $α<d$, then the sufficient condition for the existence of the solution is $d \leq 2+α$ (if $H>1/2$), respectively $d<2+α$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the "convoluted weighted" intersection local time of $k$ independent $d$-dimensional Brownian motions.