THE LAW OF THE ITERATED LOGARITHM FOR SPATIAL AVERAGES OF THE STOCHASTIC HEAT EQUATION*

doi:10.1007/s10473-023-0222-4

Abstract

Abstract: Let $u(t,x)$ be the solution to the one-dimensional nonlinear stochastic heat equation driven by space-time white noise with $u(0,x)=1$ for all $x \in \mathbb{R}$. In this paper, we prove the law of the iterated logarithm (LIL for short) and the functional LIL for a linear additive functional of the form $\int _{[0,R] } u(t,x)\mathrm{d} x$ and the nonlinear additive functionals of the form $\int_{[0, R]} g(u(t, x))\mathrm{d} x$, where $g: \mathbb{R} \rightarrow \mathbb{R}$ is nonrandom and Lipschitz continuous, as $R\rightarrow\infty$ for fixed $t>0$, using the localization argument.

Key words: law of the iterated logarithm, stochastic heat equation, Malliavin calculus

Jingyu, Li, Yong, Zhang. THE LAW OF THE ITERATED LOGARITHM FOR SPATIAL AVERAGES OF THE STOCHASTIC HEAT EQUATION^*[J].Acta mathematica scientia,Series B, 2023, 43(2): 907-918.

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THE LAW OF THE ITERATED LOGARITHM FOR SPATIAL AVERAGES OF THE STOCHASTIC HEAT EQUATION^*

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