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The Non-Linear Stochastic Wave Equation in High Dimensions


 
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1. Title Title of document The Non-Linear Stochastic Wave Equation in High Dimensions
 
2. Creator Author's name, affiliation, country Daniel Conus; Ecole Polytechnique Fédérale
 
2. Creator Author's name, affiliation, country Robert C. Dalang; Ecole Polytechnique Fédérale
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Martingale measures; stochastic integration; stochastic wave equation; stochastic partial differential equations; moment formulae; Hölder continuity
 
3. Subject Subject classification 60H15; 60H20; 60H05
 
4. Description Abstract 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Swiss National Foundation for Scientific Research
 
7. Date (YYYY-MM-DD) 2008-04-12
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/500
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-500
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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