Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition (Q2056414)

In this article, the authors study the parabolic Anderson model $$ \partial_t u(t,x) = \frac{1}{2} \partial_x^2 u(t,x) + u(t,x) \eta(t,x), \quad t > 0, \,\, x \in \mathbb{R}, $$ with initial condition $u(0) = \delta_0$, where $\eta$ denotes space-time white noise on $\mathbb{R}_+ \times \mathbb{R}$. Consider the following renormalization of the solution $$ U(t,x) := \frac{u(t,x)}{p_t(x)} \quad \text{for all }t > 0\text{ and }x \in \mathbb{R},$$ where $$ p_t(x) = \frac{1}{\sqrt{2 \pi t}} e^{- \frac{x^2}{2t}} \quad \text{for all }t > 0\text{ and }x \in \mathbb{R}.$$ In their first main result (see Theorem 1.1) the authors show that the process $U(t)$ is weakly mixing, and hence also ergodic, for every $t > 0$. Together with the ergodic theorem, it follows that, for all $t \geq 0$, $$ \lim_{N \to \infty} \mathcal{S}_{N,t} = 0\text{ a.s. and in }L^1(\Omega),$$ where $$ \mathcal{S}_{N,t} := \frac{1}{N} \int_0^N [U(t,x) - 1] \text{d}x \text{ for all }N > 0\text{ and }t \geq 0.$$ In their second main result (see Theorem 1.2) the authors provide a quantitative central limit theorem. Namely, for every $t > 0$ there exist $c = c(t) > 0$ and $N_0 = N_0(t) > \text{e}$ such that $$ d_{\mathrm{TV}} \bigg( \frac{\mathcal{S}_{N,t}}{\sqrt{\mathrm{Var}(\mathcal{S}_{N,t})}}, \mathrm{N}(0,1) \bigg) \leq c \sqrt{\frac{\log N}{N}} \text{ for all }N \geq N_0.$$ This result implies that, for all $t > 0$, $$ \sqrt{\frac{N}{\log N}} \mathcal{S}_{N,t} \overset{\text{d}}{\longrightarrow} \mathrm{N}(0,2t)\text{ as }N\to \infty.$$ More generally, in their third main result (see Theorem 1.3) the authors provide a functional central limit theorem. Namely, for any $T > 0$ we have $$ \sqrt{\frac{N}{\log N}} \mathcal{S}_{N,\bullet} \xrightarrow{C[0,T]} \sqrt{2} B\text{ as }N\to \infty,$$ where $B$ denotes a standard Brownian motion. It is left as an open problem whether the weak convergence $\xrightarrow{C[0,T]}$ in the Banach space $C[0,T]$ of continuous functions can be replaced by convergence in total variation.