SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS

doi:10.1007/s10473-019-0315-2

摘要/Abstract

摘要： 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.

关键词: Gaussian 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

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: Gaussian 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

中图分类号:

60G15

胡耀忠. SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS[J]. 数学物理学报(英文版), 2019, 39(3): 874-914.

Yaozhong HU. SOME RECENT PROGRESS ON STOCHASTIC HEAT EQUATIONS[J]. Acta mathematica scientia,Series B, 2019, 39(3): 874-914.

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