Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition

Abstract

Let $\{u(t,x){\}}_{t>0,x\in \mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition ${\delta }_0$ and driven by space-time white noise on ${\mathbb{R}}_{+}\times \mathbb{R}$, and let $p_t(x):=(2\pi t{)}^{-1/2}\exp \{-x^2/(2t)\}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers \cite{CKNP,CKNP_b} in order to prove that the random field $x\mapsto u(t,x)/p_t(x)$ is ergodic for every $t>0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari \cite{HNV2018}.

An error in the proof of Lemma 5.4 has been corrected

References in corpus (3)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Central limit theorems for stochastic wave equations in dimensions one and two
• Spatial ergodicity for SPDEs via a Poincaré-type inequality

Cited by in corpus (3)

• Almost sure central limit theorem for the hyperbolic Anderson model with Lévy white noise
• A CLT for dependent random variables, with an application to an infinite system of interacting diffusion processes
• Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\ge 1$