Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise ut = 1 2 ∆u + uẆ in R+ × R , whose solution is interpreted in the mild sense. The noise Ẇ 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 Itô-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 < d, then the sufficient condition for the existence of the solution is d ≤ 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. MSC 2000 subject classification: Primary 60H15; secondary 60H05