14. chen.kim:19:nonlinear#
Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency
Le Chen and Kunwoo Kim
Abstract: In this paper, we study the nonlinear stochastic heat equation in the spatial domain \(\mathbb{R}^d\) 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 \(\mathbb{Z}^d\) to that on \(\mathbb{R}^d\). Then we prove a localization result for the intermittency fronts, which extends results by Conus and Khoshnevisan [CK12] from one space dimension to higher space dimension. The linear case has been recently proved by Huang et al [HLN17b] using different techniques.
MSC 2010 subject classifications: Primary 60H15. Secondary 60G60, 35R60.
Keywords: stochastic heat equation, moment estimates, phase transition, intermittency, intermittency front, measure-valued initial data, moment Lyapunov exponents.
[CK19] Chen, Le & Kim, Kunwoo (2019) ‘Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency’, Acta Math. Sci. Ser. B (Engl. Ed.) 39, 645–668. <https://doi.org/10.1007/s10473-019-0303-6>
@article{chen.kim:19:nonlinear,
author = {Chen, Le and Kim, Kunwoo},
title = {Nonlinear stochastic heat equation driven by spatially colored noise: moments and intermittency},
journal = {Acta Math. Sci. Ser. B (Engl. Ed.)},
fjournal = {Acta Mathematica Scientia. Series B. English Edition},
volume = {39},
year = {2019},
number = {3},
pages = {645--668},
issn = {0252-9602},
mrclass = {60H15 (35K15 35K58 35R60 60G60)},
mrnumber = {4066498},
doi = {10.1007/s10473-019-0303-6},
url = {https://doi.org/10.1007/s10473-019-0303-6}
}
References: Balan and Chen [BC18]; Carmona and Molchanov [CM94]; Chen [Che13]; Chen and Dalang [CD15b]; Chen et al. [CHN17]; Chen and Huang [CH19a]; Chen and Kim [CK17]; Chen [Che17b]; Conus and Khoshnevisan [CK12]; Dalang [Dal99]; Dalang and Frangos [DF98]; Foondun and Khoshnevisan [FK09]; Foondun and Khoshnevisan [FK13]; Foondun and Khoshnevisan [FK14]; Foondun et al. [FLO17]; Huang [Hua17]; Huang et al. [HLN17b]; Khoshnevisan [Kho14]; Khoshnevisan and Kim [KK15]; Mueller [Mue91b]; Noble [Nob97]; Podlubny [Pod99]; Stein [Ste70]; Tessitore and Zabczyk [TZ98b]; Walsh [Wal86];