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We introduce a stochastic partial differential equation (SPDE) with ellipticoperator in divergence form, with measurable and bounded coefficients anddriven by space-time white noise. Such SPDEs could be used in mathematicalmodelling of diffusion phenomena in medium consisting of different kinds ofmaterials and undergoing stochastic perturbations. We characterize the solutionand, using the Stein--Malliavin calculus, we prove that the sequence of itsrecentered and renormalized spatial quadratic variations satisfies an almostsure central limit theorem. Particular focus is given to the interesting casewhere the coefficients of the operator are piecewise constant.

Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator