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

Yaozhong Hu (1) , Jingyu Huang (1) , David Nualart (1) , Samy Tindel (2)

1 Department of Mathematics [Lawrence, Kansas]
2 BIGS - Biology, genetics and statistics

This paper studies the stochastic heat equation with multiplicative noises of the form $u\dot{W}$, where $W$ is a mean zero Gaussian noise and the differential element $u\dot{W}$ 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.

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

Domaines: Probabilités [math.PR]

Yaozhong Hu, Jingyu Huang, David Nualart, Samy Tindel. Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electronic Journal of Probability, 2015, 20, pp.1-50. ⟨10.1214/EJP.v20-3316⟩. ⟨hal-00944904⟩