Electronic Journal of Probability
- Electron. J. Probab.
- Volume 4 (1999), paper no. 6, 29 pp.
Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s
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.
Electron. J. Probab. Volume 4 (1999), paper no. 6, 29 pp.
Accepted: 24 March 1999
First available in Project Euclid: 4 March 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 60H05: Stochastic integrals 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35D10
This work is licensed under a Creative Commons Attribution 3.0 License.
Dalang, Robert. Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s. Electron. J. Probab. 4 (1999), paper no. 6, 29 pp. doi:10.1214/EJP.v4-43. http://projecteuclid.org/euclid.ejp/1457125515.