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Acta Mathematica Scientia

, Volume 39, Issue 3, pp 874–914 | Cite as

Some Recent Progress on Stochastic Heat Equations

  • Yaozhong HuEmail author
Article
  • 88 Downloads

Abstract

This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution; Feynman-Kac formula for the moments of the solution; and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.

Key words

random field Gaussian noise stochastic partial differential equation (stochastic heat equation) Feynman-Kac formula for the solution Feynman-Kac formula for the moments of the solution chaos expansion hypercontractivity moment bounds Hölder continuity joint Hölder continuity asymptotic behaviour Trotter-Lie formula Skorohod integral 

2010 MR Subject Classification

60G15 60G22 60H05 60H07 60H10 60H15 28C20 35K15 35R60 

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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

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