Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solution and analysis of its density

In this paper we study a fractional stochastic heat equation on Rd(d≥1) with additive noise ∂∂tu(t,x)=dδ_α_u(t,x)+b(u(t,x))+W˙H(t,x) where dδ_α_ is a nonlocal fractional differential operator and W˙H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d=1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.