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

doi:10.1007/s10473-023-0222-4

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 907-918.doi: 10.1007/s10473-023-0222-4

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

Jingyu, Li, Yong, Zhang^†

School of Mathematics, Jilin University, Changchun 130012, China

收稿日期:2021-11-16 修回日期:2022-08-03 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Yong Zhang, E-mail: zyong2661@jlu.edu.cn.
作者简介:Jingyu Li, E-mail: jingyul20@mails.jlu.edu.cn
基金资助:
This work was supported by the National Natural Science Foundation of China (11771178 and 12171198), the Science and Technology Development Program of Jilin Province (20210101467JC), the Science and Technology Program of Jilin Educational Department during the "13th Five-Year" Plan Period (JJKH20200951KJ) and the Fundamental Research Funds for the Central Universities.

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

Jingyu, Li, Yong, Zhang^†

School of Mathematics, Jilin University, Changchun 130012, China

Received:2021-11-16 Revised:2022-08-03 Online:2023-03-25 Published:2023-04-12
Contact: †Yong Zhang, E-mail: zyong2661@jlu.edu.cn.
About author:Jingyu Li, E-mail: jingyul20@mails.jlu.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (11771178 and 12171198), the Science and Technology Development Program of Jilin Province (20210101467JC), the Science and Technology Program of Jilin Educational Department during the "13th Five-Year" Plan Period (JJKH20200951KJ) and the Fundamental Research Funds for the Central Universities.

摘要/Abstract

摘要： 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.

关键词: law of the iterated logarithm, stochastic heat equation, Malliavin calculus

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]. 数学物理学报(英文版), 2023, 43(2): 907-918.

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^*

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

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