23. chen.khoshnevisan.ea:21:spatial#
Spatial ergodicity for SPDEs via Poincaré-type inequalities
Le Chen, Davar Khoshnevisan, David Nualart, and Fei Pu
Abstract: Consider a parabolic stochastic PDE of the form
where \(u=u(t\,,x)\), \(t\ge 0\), \(x\in\mathbb{R}^d\), \(\sigma:\mathbb{R}\to\mathbb{R}\) is Lipschitz continuous and non random, and \(\eta\) is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation \(f\). If, in addition, \(u(0)\equiv1\), then we prove that, under a mild decay condition on \(f\), the process \(x\mapsto u(t\,,x)\) is stationary and ergodic at all times \(t>0\). It has been argued that, when coupled with moment estimates, spatial ergodicity of \(u\) teaches us about the intermittent nature of the solution to such SPDEs [BC95a, Kho14]. Our results provide rigorous justification of such discussions.
Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field \(u(t)\) is mixing for every \(t>0\); and by (b) giving a quick proof of a conjecture of Conus et al [CJK12] about the ‘’size’’ of the intermittency islands of \(u\).
The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama [Mar49] (see also Dym and McKean [DM76]) in the simple setting where the nonlinear term \(\sigma\) is a constant function.
MSC 2010 subject classifications: Primary. 60H15; Secondary. 37A25, 60H07, 60G10.
Keywords: SPDEs, ergodicity, Malliavin calculus, Poincaré-type inequality.
[CKNP21b] Chen, Le, Khoshnevisan, Davar, Nualart, David & Pu, Fei (2021) ‘Spatial ergodicity for SPDEs via Poincar’e-type inequalities’, Electron. J. Probab. 26, Paper No. 140, 37. <https://doi.org/10.1214/21-ejp690>
@article{chen.khoshnevisan.ea:21:spatial,
author = {Chen, Le and Khoshnevisan, Davar and Nualart, David and Pu, Fei},
title = {Spatial ergodicity for {SPDE}s via {P}oincar\'{e}-type inequalities},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {26},
year = {2021},
pages = {Paper No. 140, 37},
mrclass = {60H15 (37A25 60G10 60H07)},
mrnumber = {4346664},
doi = {10.1214/21-ejp690},
url = {https://doi.org/10.1214/21-ejp690}
}
References: Bertini and Cancrini [BC95a]; Burkholder [Bur66]; Burkholder et al. [BDG72]; Burkholder and Gundy [BG70]; Capitaine et al. [CHL97]; Cardon-Weber and Millet [CWM04]; Carlen and Krée [CK91]; Chen et al. [CHN17]; Chen and Huang [CH19b]; Chen [Che16b]; Chen et al. [CHNT17]; Conus et al. [CJK12]; Conus et al. [CJK13]; Conus and Khoshnevisan [CK12]; Conus et al. [CJKS13b]; Dalang [Dal99]; Dalang and Frangos [DF98]; Davis [Dav76]; Doob [Doo90]; Dym and McKean [DM76]; Edgar and Sucheston [ES92]; Foondun and Khoshnevisan [FK09]; Foondun and Khoshnevisan [FK13]; Gaveau and Trauber [GT82]; Hawkes [Haw79]; Hawkes [Haw84]; Hu et al. [HHNT15]; Hu et al. [HHL+17]; Huang et al. [HLN17b]; Huang et al. [HNV20]; Huang et al. [HNVZ20]; Kahane [Kah85]; Karczewska and Zabczyk [KZ01]; Khoshnevisan [Kho02]; Khoshnevisan [Kho14]; Khoshnevisan et al. [KKX17]; Lépingle and Ouvrard [LO73]; Maruyama [Mar49]; Millet and Sanz-Solé [MSS99]; Mueller [Mue91b]; Mueller [Mue09]; Nualart [Nua09]; Ocone [Oco84]; Olver et al. [OLBC10]; Ouvrard [Ouv76]; Peszat [Pes02]; Peszat and Zabczyk [PZ00]; Stein [Ste70]; Walsh [Wal86];