Central limit theorems for heat equation with time-independent noise: the regular and rough cases

Abstract

In this article, we investigate the asymptotic behaviour of the spatial integral of the solution to the parabolic Anderson model with time independent noise in dimension d ≥ 1, as the domain of the integral becomes large. We consider 3 cases: (a) the case when the noise has an integrable covariance function; (b) the case when the covariance of the noise is given by the Riesz kernel; (c) the case of the rough noise, i.e. fractional noise with index H ∈ ( 1 4 , 1 2 ) in dimension d = 1. In each case, we identify the order of magnitude of the variance of the spatial integral, we prove a quantitative central limit theorem for the normalized spatial integral by estimating its total variation distance to a standard normal distribution, and we give the corresponding functional limit result.

Quantitative central limit theorems for the parabolic Anderson model driven by colored noises

“…(iii) Some recent results related to the spatial average of parabolic and hyperbolic Anderson model with time-independent noises can be found in e.g. [2,3].…”

“…(i-b) in case (3), that is when the spatial correlation is given by that of a fractional Brownian noise with Hurst index H 1 < 1 2 (see Hypothesis 2), it has been pointed out in [30,Page 910] that due to the fact that the associated spectral density vanishes at zero, the first chaotic component of F R (t) is asymptotically negligible compared to other chaoses (each of them has limiting variance of order R), then the rate O(R −1/2 ) is consistent with the CLT heuristic.…”