Journal of Theoretical Probability

, Volume 27, Issue 2, pp 449–492 | Cite as

Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

  • Pierre A. Vuillermot
  • Jean C. Zambrini


In this article we prove the existence of Bernstein processes which we associate in a natural way with a class of non-autonomous linear parabolic initial- and final-boundary value problems defined in bounded convex subsets of Euclidean space of arbitrary dimension. Under certain conditions regarding their joint endpoint distributions, we also prove that such processes become reversible Markov diffusions. Furthermore we show that those diffusions satisfy two Itô equations for some suitably constructed Wiener processes, and from that analysis derive Feynman–Kac representations for the solutions to the given equations. We then illustrate some of our results by considering the heat equation with Neumann boundary conditions both in a one-dimensional bounded interval and in a two-dimensional disk.


Diffusion processes Parabolic partial differential equations 

Mathematics Subject Classification

35K20 60H30 60K99 



The research of both authors was supported by the FCT of the Portuguese government under Grant PTDC/MAT/69635/2006, and by the Mathematical Physics Group of the University of Lisbon under Grant ISFL/1/208. The first author is also indebted to Madalina Deaconu and Elton Hsu for stimulating discussions and correspondence on the theme of reflected diffusions. Last but not least, he wishes to thank the Complexo Interdisciplinar da Universidade de Lisboa and the ETH-Forschungsinstitut für Mathematik in Zurich where parts of this work were completed for their financial support and warm hospitality.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.UMR-CNRS 7502Institut Élie CartanNancyFrance
  2. 2.Grupo de Física Matemática da Universidade de LisboaLisbonPortugal

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