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Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s

   
@article{EJP43,
	author = {Robert Dalang},
	title = {Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {4},
	year = {1999},
	keywords = {stochastic wave equation, stochastic heat equation,Gaussian noise, process solution.},
	abstract = {We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation in dimensions greater than two. Even when the integrand is a distribution, the value of our stochastic integral process is a real-valued martingale. We use this extended integral to recover necessary and sufficient conditions under which the linear wave equation driven by spatially homogeneous Gaussian noise has a process solution, and this in any spatial dimension. Under this condition, the non-linear three dimensional wave equation has a global solution. The same methods apply to the damped wave equation, to the heat equation and to various parabolic equations.},
	pages = {no. 6, 1-29},
	issn = {1083-6489},
	doi = {10.1214/EJP.v4-43},    
        url = {http://ejp.ejpecp.org/article/view/43}}