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Acta Mathematica Scientia

, Volume 39, Issue 3, pp 669–690 | Cite as

On the Necessary and Sufficient Conditions to Solve A Heat Equation with General Additive Gaussian Noise

  • Yaozhong HuEmail author
  • Yanghui LiuEmail author
  • Samy TindelEmail author
Article
  • 34 Downloads

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

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

2010 MR Subject Classification

60G15 60H07 60H15 

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Copyright information

© Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences 2019

Authors and Affiliations

  1. 1.Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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