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Consider the parabolic Anderson model of the form, wherefor  t &gt;0 andwith, and  η  is a centered Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure  f  that satisfies Dalang's condition. Letdenote the standard Gaussian heat kernel onand setfor all  t &gt;0 and. In this paper, we present an almost sure central limit theorem (ASCLT) and a functional ASCLT for spatial averages of the formasfor fixed  t &gt;0 based on the quantitative analysis of  f . In particular, when  f  is given by a Riesz kernel, that is,for some, we can also obtain the ASCLT.

An almost sure central limit theorem for the parabolic Anderson model with delta initial condition