Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension d≥ 1

2020/07/04 by Khoshnevisan, Davar, Nualart, David, Pu, Fei · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2007.01987

Abstract

Suppose that \u(t , x)\_{t >0, x ∈ℝ^d} is the solution to a d-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure f which satisfies Dalang's condition. Let \boldsymbolp_t(x):=(2πt)^-d/2exp\-‖x‖²/(2t)\ denote the standard Gaussian heat kernel on ℝ^d. We prove that for all t>0, the process U(t):=\u(t , x)/\boldsymbolp_t(x): x∈ ℝ^d\ is stationary using Feynman-Kac's formula, and is ergodic under the additional condition f\0\=0, where f is the Fourier transform of f. Moreover, using Malliavin-Stein method, we investigate various central limit theorems for U(t) based on the quantitative analysis of f. In particular, when f is given by Riesz kernel, i.e., f(d x) = ‖x‖^-βd x, we obtain a multiple phase transition for the CLT for U(t) from β∈(0 ,1) to β=1 to β∈(1 ,d\wedge 2).

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