Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Abstract

We consider the stochastic heat equation with multiplicative noise \(u_{t}=\frac{1}{2}\Delta u+u\dot{W}\) in ℝ+×ℝd, whose solution is interpreted in the mild sense. The noise \(\dot{W}\) is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Ito-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α 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 kth 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.