NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY

doi:10.1007/s10473-019-0303-6

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (3): 645-668.doi: 10.1007/s10473-019-0303-6

NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY

陈乐¹, Kunwoo KIM²

1. Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 S. Maryland Pkwy, Las Vegas, Nevada, 89154-4020, USA;
2. Department of Mathematics, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, 37673, Korea

收稿日期:2018-02-08 出版日期:2019-06-25 发布日期:2019-06-27
作者简介:Le CHEN,E-mail:le.chen@unlv.edu;Kunwoo KIM,E-mail:kunwoo@postech.ac.kr
基金资助:
The second author is supported by the National Research Foundation of Korea (NRF-2017R1C1B1005436) and the TJ Park Science Fellowship of POSCO TJ Park Foundation.

NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY

Le CHEN¹, Kunwoo KIM²

1. Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 S. Maryland Pkwy, Las Vegas, Nevada, 89154-4020, USA;
2. Department of Mathematics, Pohang University of Science and Technology, 77 Cheongam-Ro, Nam-Gu, Pohang, Gyeongbuk, 37673, Korea

Received:2018-02-08 Online:2019-06-25 Published:2019-06-27
Supported by:
The second author is supported by the National Research Foundation of Korea (NRF-2017R1C1B1005436) and the TJ Park Science Fellowship of POSCO TJ Park Foundation.

摘要/Abstract

摘要： In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z^d to that on R^d. Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan[9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al[17] using different techniques.

关键词: Stochastic heat equation, moment estimates, phase transition, intermittency, intermittency front, measure-valued initial data, moment Lyapunov exponents

Abstract: In this article, we study the nonlinear stochastic heat equation in the spatial domain R^d subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and nonnegative-definite function that satisfies Dalang's condition. We establish the existence and uniqueness of a random field solution starting from measure-valued initial data. We find the upper and lower bounds for the second moment. With these moment bounds, we first establish some necessary and sufficient conditions for the phase transition of the moment Lyapunov exponents, which extends the classical results from the stochastic heat equation on Z^d to that on R^d. Then, we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan[9] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al[17] using different techniques.

Key words: Stochastic heat equation, moment estimates, phase transition, intermittency, intermittency front, measure-valued initial data, moment Lyapunov exponents

中图分类号:

60H15

陈乐, Kunwoo KIM. NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY[J]. 数学物理学报(英文版), 2019, 39(3): 645-668.

Le CHEN, Kunwoo KIM. NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY[J]. Acta mathematica scientia,Series B, 2019, 39(3): 645-668.

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NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY

NONLINEAR STOCHASTIC HEAT EQUATION DRIVEN BY SPATIALLY COLORED NOISE: MOMENTS AND INTERMITTENCY

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