JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

doi:10.1007/s10473-019-0309-0

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (3): 764-780.doi: 10.1007/s10473-019-0309-0

JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

胡耀忠¹, Khoa LÊ²

1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
2. Department of Mathematics, South Kensington Campus, Imperial College London, London, SW7 2AZ, United Kingdom

收稿日期:2018-03-04 修回日期:2019-03-03 出版日期:2019-06-25 发布日期:2019-06-27
作者简介:Yaozhong HU,E-mail:yaozhong@ualberta.ca;Khoa LÊ,E-mail:n.le@imperial.ac.uk
基金资助:
Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; K. Lê is supported by Martin Hairer's Leverhulme Trust leadership award.

JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

Yaozhong HU¹, Khoa LÊ²

1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
2. Department of Mathematics, South Kensington Campus, Imperial College London, London, SW7 2AZ, United Kingdom

Received:2018-03-04 Revised:2019-03-03 Online:2019-06-25 Published:2019-06-27
Supported by:
Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; K. Lê is supported by Martin Hairer's Leverhulme Trust leadership award.

摘要/Abstract

摘要： We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation (_t-1/2△)u=u◇W in d-dimensional space, where W is a mean zero Gaussian noise with temporal covariance γ₀ and spatial covariance given by a spectral density μ(ξ). We assume that γ₀(t) ≤ c|t|^-α₀ and |μ(ξ)| ≤ c|ξ_i|^-α_i or |μ(ξ)| ≤ c|ξ|^-α, where α_i, i=1,…, d (or α) can take negative value.

关键词: Gaussian process, stochastic heat equation, parabolic Anderson model, multiplicative noise, chaos expansion, hypercontractivity, Hölder continuity, joint Hölder continuity

Abstract: We obtain the Holder continuity and joint Holder continuity in space and time for the random field solution to the parabolic Anderson equation (_t-1/2△)u=u◇W in d-dimensional space, where W is a mean zero Gaussian noise with temporal covariance γ₀ and spatial covariance given by a spectral density μ(ξ). We assume that γ₀(t) ≤ c|t|^-α₀ and |μ(ξ)| ≤ c|ξ_i|^-α_i or |μ(ξ)| ≤ c|ξ|^-α, where α_i, i=1,…, d (or α) can take negative value.

Key words: Gaussian process, stochastic heat equation, parabolic Anderson model, multiplicative noise, chaos expansion, hypercontractivity, Hölder continuity, joint Hölder continuity

中图分类号:

60H15

胡耀忠, Khoa LÊ. JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL[J]. 数学物理学报(英文版), 2019, 39(3): 764-780.

Yaozhong HU, Khoa LÊ. JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL[J]. Acta mathematica scientia,Series B, 2019, 39(3): 764-780.

参考文献

[1] Balan R, Quer-Sardanyons L, Song J. Hölder continuity for the Parabolic Anderson Model with spacetime homogeneous Gaussian noise. Acta Mathematica Scientia, 2019, 39B(3):717-730. See also arXiv preprint. 1807.05420
[2] Chen L, Dalang R C. Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions. Stoch Partial Differ Equ Anal Comput, 2014, 2(3):316-352
[3] Chen L, Kalbasi K, Hu Y, Nualart D. Intermittency for the stochastic heat equation driven by timefractional Gaussian noise with H ∈ (0, 1/2). Prob Theory and Related Fields, 2018, 171(1/2):431-457
[4] Hu Y. Analysis on Gaussian space. Singapore:World Scientific, 2017
[5] Hu Y, Le K. A multiparameter Garsia-Rodemich-Rumsey inequality and some applications. Stochastic Process Appl, 2013, 123(9):3359-3377
[6] Hu Y, Nualart D. Stochastic heat equation driven by fractional noise and local time. Probab Theory Related Fields, 2009, 143(1/2):285-328
[7] Hu Y, Huang J, Lê K, Nualart D, Tindel S. Stochastic heat equation with rough dependence in space. Ann Probab, 2017, 45(6B):4561-4616
[8] Hu Y, Huang J, Lê K, Nualart D, Tindel S. Parabolic Anderson model with rough dependence in space. Computation and Combinatorics in Dynamics, Stochastics and Control, 2018:477-498
[9] Hu Y, Huang J, Nualart D, Tindel S. Stochastic heat equations with general multiplicative Gaussian noises:Hölder continuity and intermittency. Electron J Probab, 2015, 20(55):50 pp
[10] Hu Y, Nualart D, Song J. A nonlinear stochastic heat equation:Hölder continuity and smoothness of the density of the solution. Stochastic Process Appl, 2013, 123(3):1083-1103
[11] Hu Y, Nualart D, Song J. Feynman-Kac formula for heat equation driven by fractional white noise. Ann Probab, 2011, 30:291-326
[12] Memin J, Mishura Y, Valkeila E. Inequalities for the moments of Wiener integrals with respect to a fractional Brownian motion. Statist Probab Lett, 2001, 51:197-206
[13] Nualart D. The Malliavin calculus and related topics. Second Edition. Probability and its Applications (New York). Berlin:Springer-Verlag, 2006. xiv+382 pp
[14] Sanz-Solé M, Sarrà M. Hölder continuity for the stochastic heat equation with spatially correlated noise//Seminar on Stochastic Analysis, Random Fields and Applications, Ⅲ (Ascona, 1999), 259-268, Progr Probab, 52. Basel:Birkhäuser, 2002

JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

JOINT HÖLDER CONTINUITY OF PARABOLIC ANDERSON MODEL

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