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Limit theorems for time-dependent averages of nonlinear stochastic heat equations. (English) Zbl 1482.60086

Summary: 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 \(u(t,x)\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
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