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We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein-Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the interesting case where the coefficients of the operator are piecewise constant.Keywords Stochastic partial differential equations, divergence form, piecewise constant coefficients, fundamental solution, Stein-Malliavin calculus, almost sure central limit theorem 2010 MSC 60H15, 60G15, 60H05, 35A08

We introduce a fractional stochastic heat equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by an infinite-dimensional fractional Brownian motion. We characterize the fundamental solution of its deterministic part, and prove the existence and the uniqueness of its solution.Here the coefficients A and ρ have the following form A(x) = a 1 1 {x≤0} + a 2 1 {x>0} and ρ(x) = ρ 1 1 {x≤0} + ρ 2 1 {x>0} , 2010 Mathematics Subject Classification. 60G22, 60H15, 35R60.

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