Limit theorems for time-dependent averages of nonlinear stochastic heat equations

Abstract

We study limit theorems for time-dependent averages of the form $X_t:=\frac{1}{2L(t)}{\int }_{-L(t)}^{L(t)}u(t,x)dx$, as $t\to \infty$, where $L(t)=\exp (\lambda t)$ and $u(t,x)$ is the solution to a stochastic heat equation on ${\mathbb{R}}_{+}\times \mathbb{R}$ driven by space-time white noise with $u_0(x)=1$ for all $x\in \mathbb{R}$. We show that for $X_t$ (i) the weak law of large numbers holds when $\lambda >{\lambda }_1$, (ii) the strong law of large numbers holds when $\lambda >{\lambda }_2$, (iii) the central limit theorem holds when $\lambda >{\lambda }_3$, but fails when $\lambda <{\lambda }_4\le {\lambda }_3$, (iv) the quantitative central limit theorem holds when $\lambda >{\lambda }_5$, where ${\lambda }_i$'s are positive constants depending on the moment Lyapunov exponents of $u(t,x)$.

25 pages