In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space–time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius R converges, as R tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

In this article we present a quantitative central limit theorem for the stochastic fractional heat equation driven by a a general Gaussian multiplicative noise, including the cases of space-time white noise and the white-colored noise with spatial covariance given by the Riesz kernel or a bounded integrable function. We show that the spatial average over a ball of radius R converges, as R tends to infinity, after suitable renormalization, towards a Gaussian limit in the total variation distance. We also provide a functional central limit theorem. As such, we extend recently proved similar results for stochastic heat equation to the case of the fractional Laplacian and to the case of general noise.

Via Malliavin calculus, we analyze the limit behavior in distribution of the spatial wavelet variation for the solution to the stochastic linear wave equation with fractional Gaussian noise in time and white noise in space. We propose a wavelet-type estimator for the Hurst parameter of the this solution and we study its asymptotic properties.

No abstract