Electronic Journal of Probability

Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

Yaozhong Hu, Jingyu Huang, David Nualart, and Samy Tindel

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This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 55, 50 pp.

Accepted: 23 May 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60H10: Stochastic ordinary differential equations [See also 34F05] 65C30: Stochastic differential and integral equations

Fractional Brownian motion Malliavin calculus Skorohod integral Young's integral stochastic partial differential equations Feynman-Kac formula intermittency

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Hu, Yaozhong; Huang, Jingyu; Nualart, David; Tindel, Samy. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20 (2015), paper no. 55, 50 pp. doi:10.1214/EJP.v20-3316. https://projecteuclid.org/euclid.ejp/1465067161

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