Abstract
We illustrate the use of white noise analysis in the solution of stochastic partial differential equations by explicitly solving the stochastic Neumann boundary-value problem LU(x)−c(x)U(x)=0, x∈D⊂R d,γ(x)⋅∇U(x)=−W(x), x∈∂D, where L is a uniformly elliptic linear partial differential operator and W(x), x∈R d, is d-parameter white noise.
Similar content being viewed by others
References
Bass, R.: Diffusions and Elliptic Operators, Springer, Berlin, 1998.
Duff, G. F. and Naylor, D.: Differential Equations of Applied Mathematics, Wiley, New York, 1966.
Freidlin, M.: Functional Integration and Partial Differential Equations, Princeton University Press, Princeton, 1985.
Hida, T.: Brownian Motion, Springer, Berlin, 1980.
Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L.: White Noise Analysis, Kluwer Acad. Publ., Dordrecht, 1993.
Holden, H., Øksendal, B., Ubøe, J. and Zhang, T.: Stochastic Partial Differential Equations, Birkhäuser, Basel, 1996.
Kuo, H.-H.: White Noise Distribution Theory, CRC Press, Boca Raton, 1996.
Oberguggenberger, M. and Russo, F.: Nonlinear SPDEs: Colombeau solutions and pathwise limits, In: L. Decreusefond, J. Gjerde, B. Øksendal and A. S. Ñstünel (eds), Stochastic Analysis and Related Topics VI, Birkhäuser, Boston, 1998, pp. 319–332.
Walsh, J. B.: An introduction to stochastic partial differential equations, In: R. Carmona, H. Kesten and J. B. Walsh (eds), École d'été de probabilités de Saint-Flour XIV 1984, Lecture Notes in Math. 1180, Springer, Berlin, 1984, pp. 236–433.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Holden, H., Øksendal, B. A White Noise Approach to Stochastic Neumann Boundary-Value Problems. Acta Applicandae Mathematicae 63, 141–150 (2000). https://doi.org/10.1023/A:1010730510108
Issue Date:
DOI: https://doi.org/10.1023/A:1010730510108