The Annals of Probability

Intermittency for the wave and heat equations with fractional noise in time

Raluca M. Balan and Daniel Conus

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Abstract

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.

Article information

Source
Ann. Probab., Volume 44, Number 2 (2016), 1488-1534.

Dates
Received: October 2013
Revised: June 2014
First available in Project Euclid: 14 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1457960400

Digital Object Identifier
doi:10.1214/15-AOP1005

Mathematical Reviews number (MathSciNet)
MR3474476

Zentralblatt MATH identifier
1343.60081

Subjects
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

Keywords
Stochastic heat and wave equations spatially homogeneous noise fractional Brownian motion Malliavin calculus intermittency

Citation

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


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