Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise u(t) = 1/2 Delta u + u(W) over dot in R+ x R-d, whose solution is interpreted in the mild sense. The noise (W) over dot 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 alpha, or the Bessel kernel of order alpha < d, then the sufficient condition for the existence of the solution is d <= 2 + alpha (if H > 1/2), respectively d < 2 + alpha (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.