HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE

doi:10.1007/s10473-019-0306-3

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (3): 717-730.doi: 10.1007/s10473-019-0306-3

HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE

Raluca M BALAN¹, Lluís QUER-SARDANYONS², 宋健³

1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada;
2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra(Barcelona), Catalonia, Spain;
3. School of Mathematics, Shandong University, Jinan 250100, China

收稿日期:2018-07-18 修回日期:2019-03-04 出版日期:2019-06-25 发布日期:2019-06-27
作者简介:Raluca M BALAN,E-mail:rbalan@uottawa.ca;Lluís QUER-SARDANYONS,E-mail:quer@mat.uab.cat;Jian SONG,E-mail:txjsong@hotmail.com
基金资助:
The first author is supported by a grant from the Natural Sciences and Engineering Research Council of Canada, and the second author is supported by the grant MTM2015-67802P.

HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE

Raluca M BALAN¹, Lluís QUER-SARDANYONS², Jian SONG³

1. Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada;
2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra(Barcelona), Catalonia, Spain;
3. School of Mathematics, Shandong University, Jinan 250100, China

Received:2018-07-18 Revised:2019-03-04 Online:2019-06-25 Published:2019-06-27
Supported by:
The first author is supported by a grant from the Natural Sciences and Engineering Research Council of Canada, and the second author is supported by the grant MTM2015-67802P.

摘要/Abstract

摘要： In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in L^p(Ω). Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in [6, 10] under more restrictive conditions, and with sub-optimal exponents for H¨older continuity.

关键词: Gaussian noise, stochastic partial differential equations, Malliavin calculus

Abstract: In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in L^p(Ω). Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in [6, 10] under more restrictive conditions, and with sub-optimal exponents for H¨older continuity.

Key words: Gaussian noise, stochastic partial differential equations, Malliavin calculus

中图分类号:

60H15

Raluca M BALAN, Lluís QUER-SARDANYONS, 宋健. HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE[J]. 数学物理学报(英文版), 2019, 39(3): 717-730.

Raluca M BALAN, Lluís QUER-SARDANYONS, Jian SONG. HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE[J]. Acta mathematica scientia,Series B, 2019, 39(3): 717-730.

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HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE

HÖLDER CONTINUITY FOR THE PARABOLIC ANDERSON MODEL WITH SPACE-TIME HOMOGENEOUS GAUSSIAN NOISE

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