The Annals of Probability
- Ann. Probab.
- Volume 44, Number 2 (2016), 1488-1534.
Intermittency for the wave and heat equations with fractional noise in time
In this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index $H>1/2$. The solutions of these equations are interpreted in the Skorohod sense. Using Malliavin calculus techniques, we obtain an upper bound for the moments of order $p\geq2$ of the solution. In the case of the wave equation, we derive a Feynman–Kac-type formula for the second moment of the solution, based on the points of a planar Poisson process. This is an extension of the formula given by Dalang, Mueller and Tribe [Trans. Amer. Math. Soc. 360 (2008) 4681–4703], in the case $H=1/2$, and allows us to obtain a lower bound for the second moment of the solution. These results suggest that the moments of the solution grow much faster in the case of the fractional noise in time than in the case of the white noise in time.
Ann. Probab., Volume 44, Number 2 (2016), 1488-1534.
Received: October 2013
Revised: June 2014
First available in Project Euclid: 14 March 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] 60H07: Stochastic calculus of variations and the Malliavin calculus
Balan, Raluca M.; Conus, Daniel. Intermittency for the wave and heat equations with fractional noise in time. Ann. Probab. 44 (2016), no. 2, 1488--1534. doi:10.1214/15-AOP1005. https://projecteuclid.org/euclid.aop/1457960400