Summary
We consider the equationu t =u xx +u γ W forx on a finite interval, with Dirichlet boundary conditions. W is spacetime white noise. The initial condition is continuous and nonnegative. We show existence and uniqueness for all time, provided 1 ≧γ<3/2.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Dawson, D.A., Iscoe, I., Perkins, E.A.: Super-Brownian motion: path properties an hitting probabilities. Probab. Theory Relat. Fields83, 135–206 (1989)
McKean, Jr., H.P. Stochastic integrals. New York: Academic Press 1969
Mueller, C.: Limit results for two stochastic partial differential equations Stochastics (to appear)
Mueller, C.: On the support of solutions to the heat equation with noise. Stochastics (to appear)
Shiga, T.: private communication
Varadhan, S.R.S.: Large deviations and applications. SIAM, 1984
Walsh, J.B.: An introduction to stochastic partial differential equations. (Lect. Notes Math., vol. 1180) Berlin, Heidelberg New York: Springer 1986
Williams, D.: Diffusions, Markov processes, and martingales. New York: Wiley 1979
Author information
Authors and Affiliations
Additional information
Supported by an NSF grant
Rights and permissions
About this article
Cite this article
Mueller, C. Long time existence for the heat equation with a noise term. Probab. Th. Rel. Fields 90, 505–517 (1991). https://doi.org/10.1007/BF01192141
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01192141