Probabilistic representations of solutions to the heat equation
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In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if ϕ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition ϕ, is given by the convolution of ϕ with the heat kernel (Gaussian density). Our results also extend the probabilistic representation of solutions of the heat equation to initial conditions that are arbitrary tempered distributions.
KeywordsBrownian motion heat equation translation operators infinite dimensional stochastic differential equations
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- Bass Richard F, Diffusions and elliptic operators (Springer) (1998)Google Scholar
- Freidlin M, Functional integration and partial differential equations,Ann. Math. Stud. (NJ: Princeton University Press, Princeton) (1985) no. 109Google Scholar
- Gawarecki L, Mandrekar V and Rajeev B, From finite to infinite dimensional stochastic differential equations (preprint)Google Scholar
- Ito K, Foundations of stochastic differential equations in infinite dimensional spaces, Proceedings of CBMS-NSF National Conference Series in Applied Mathematics, SIAM (1984)Google Scholar
- Kallianpur G and Xiong J, Stochastic differential equations in infinite dimensional spaces, Lecture Notes,Monogr. Series (Institute of Mathematical Statistics) (1995) vol. 26Google Scholar
- Métivier M, Semi-martingales-A course on stochastic processes (Walter de Gruyter) (1982)Google Scholar
- Rajeev B, From Tanaka’s formula to Ito’s formula: Distributions, tensor products and local times, Séminaire de Probabilite’s XXXV,Lecture Notes in Math. 1755 (Springer-Verlag) (2001)Google Scholar
- Thangavelu S, Lectures on Hermite and Laguerre expansions,Math. Notes 42 (N.J: Princeton University Press, Princeton) (1993)Google Scholar