ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

doi:10.1007/s10473-019-0304-5

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (3): 669-690.doi: 10.1007/s10473-019-0304-5

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

胡耀忠¹, 刘阳辉², Samy TINDEL²

1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
2. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

收稿日期:2018-03-22 修回日期:2018-12-04 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;Yanghui LIU,E-mail:liu2048@purdue.edu;Samy TINDEL,E-mail:stindel@purdue.edu
基金资助:
Y. Hu is supported by an NSERC grant and a startup fund of University of Alberta; S. Tindel is supported by the NSF grant DMS1613163.

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

Yaozhong HU¹, Yanghui LIU², Samy TINDEL²

1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada;
2. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Received:2018-03-22 Revised:2018-12-04 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; S. Tindel is supported by the NSF grant DMS1613163.

摘要/Abstract

摘要： In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively, based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

关键词: Stochastic heat equation, general Gaussian noise, L² solution, sufficient and necessary condition, Wong-Zakai approximation, pathwise solution, Hölder continuity, Besov space

Abstract: In this note, we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively, based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

Key words: Stochastic heat equation, general Gaussian noise, L² solution, sufficient and necessary condition, Wong-Zakai approximation, pathwise solution, Hölder continuity, Besov space

中图分类号:

60G15

胡耀忠, 刘阳辉, Samy TINDEL. ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE[J]. 数学物理学报(英文版), 2019, 39(3): 669-690.

Yaozhong HU, Yanghui LIU, Samy TINDEL. ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE[J]. Acta mathematica scientia,Series B, 2019, 39(3): 669-690.

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ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

ON THE NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE A HEAT EQUATION WITH GENERAL ADDITIVE GAUSSIAN NOISE

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