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2008 The Non-Linear Stochastic Wave Equation in High Dimensions
Daniel Conus, Robert Dalang
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Electron. J. Probab. 13: 629-670 (2008). DOI: 10.1214/EJP.v13-500


We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on $p$-th moments of the solution ($p\geq 1$), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal.


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Daniel Conus. Robert Dalang. "The Non-Linear Stochastic Wave Equation in High Dimensions." Electron. J. Probab. 13 629 - 670, 2008.


Accepted: 12 April 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1187.60049
MathSciNet: MR2399293
Digital Object Identifier: 10.1214/EJP.v13-500

Primary: 60H15
Secondary: 60H05 , 60H20

Keywords: Hölder continuity , martingale measures , moment formulae , stochastic integration , Stochastic partial differential equations , Stochastic wave equation

Vol.13 • 2008
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