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March 2016 Intermittency for the wave and heat equations with fractional noise in time
Raluca M. Balan, Daniel Conus
Ann. Probab. 44(2): 1488-1534 (March 2016). DOI: 10.1214/15-AOP1005


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.


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Raluca M. Balan. Daniel Conus. "Intermittency for the wave and heat equations with fractional noise in time." Ann. Probab. 44 (2) 1488 - 1534, March 2016.


Received: 1 October 2013; Revised: 1 June 2014; Published: March 2016
First available in Project Euclid: 14 March 2016

zbMATH: 1343.60081
MathSciNet: MR3474476
Digital Object Identifier: 10.1214/15-AOP1005

Primary: 60H15
Secondary: 37H15 , 60H07

Keywords: fractional Brownian motion , Intermittency , Malliavin calculus , spatially homogeneous noise , Stochastic heat and wave equations

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 2 • March 2016
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