Electronic Journal of Probability

The Non-Linear Stochastic Wave Equation in High Dimensions

Daniel Conus and Robert Dalang

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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.

Article information

Electron. J. Probab. Volume 13 (2008), paper no. 22, 629-670.

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

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Mathematical Reviews number (MathSciNet)

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H20: Stochastic integral equations 60H05: Stochastic integrals

Martingale measures stochastic integration stochastic wave equation stochastic partial differential equations moment formulae Hölder continuity

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Conus, Daniel; Dalang, Robert. The Non-Linear Stochastic Wave Equation in High Dimensions. Electron. J. Probab. 13 (2008), paper no. 22, 629--670. doi:10.1214/EJP.v13-500. http://projecteuclid.org/euclid.ejp/1464819095.

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