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

Abstract

We study limit theorems for time-dependent averages of the form ${\mathit{X}}_{\mathit{t}}:=\frac{1}{2\mathit{L}(\mathit{t})}{\int }_{-\mathit{L}(\mathit{t})}^{\mathit{L}(\mathit{t})}\mathit{u}(\mathit{t},\mathit{x})\mathit{d}\mathit{x}$, as $\mathit{t}\to \infty$, where $\mathit{L}(\mathit{t})=exp(\mathit{\lambda }\mathit{t})$ and $\mathit{u}(\mathit{t},\mathit{x})$ is the solution to a stochastic heat equation on ${ℝ}_{+}\times ℝ$ driven by space-time white noise with ${\mathit{u}}_0(\mathit{x})=1$ for all $\mathit{x}\in ℝ$. We show that for ${\mathit{X}}_{\mathit{t}}$

• the weak law of large numbers holds when $\mathit{\lambda }>{\mathit{\lambda }}_1$,

• the strong law of large numbers holds when $\mathit{\lambda }>{\mathit{\lambda }}_2$,

• the central limit theorem holds when $\mathit{\lambda }>{\mathit{\lambda }}_3$, but fails when $\mathit{\lambda }<{\mathit{\lambda }}_4\le {\mathit{\lambda }}_3$,

• the quantitative central limit theorem holds when $\mathit{\lambda }>{\mathit{\lambda }}_5$,

where ${\mathit{\lambda }}_{\mathit{i}}$’s are positive constants depending on the moment Lyapunov exponents of $\mathit{u}(\mathit{t},\mathit{x})$.