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February 2022 Limit theorems for time-dependent averages of nonlinear stochastic heat equations
Kunwoo Kim, Jaeyun Yi
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Bernoulli 28(1): 214-238 (February 2022). DOI: 10.3150/21-BEJ1339

Abstract

We study limit theorems for time-dependent averages of the form Xt:=12L(t)L(t)L(t)u(t,x)dx, as t, where L(t)=exp(λt) and u(t,x) is the solution to a stochastic heat equation on +× driven by space-time white noise with u0(x)=1 for all x. We show that for Xt

  • the weak law of large numbers holds when λ>λ1,

  • the strong law of large numbers holds when λ>λ2,

  • the central limit theorem holds when λ>λ3, but fails when λ<λ4λ3,

  • the quantitative central limit theorem holds when λ>λ5,

where λi’s are positive constants depending on the moment Lyapunov exponents of u(t,x).

Funding Statement

The authors were supported by the NRF (National Research Foundation of Korea) Grants 2019R1A5A1028324 and 2020R1A2C4002077.

Acknowledgments

We appreciate Davar Khoshnevisan for stimulating discussions and suggestions. We also thank Carl Mueller and Mohammud Foondun for useful discussions.

Citation

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Kunwoo Kim. Jaeyun Yi. "Limit theorems for time-dependent averages of nonlinear stochastic heat equations." Bernoulli 28 (1) 214 - 238, February 2022. https://doi.org/10.3150/21-BEJ1339

Information

Received: 1 December 2020; Revised: 1 March 2021; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1339

Keywords: central limit theorem , Stochastic heat equation , Strong law of large numbers , Weak law of large numbers

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 1 • February 2022
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