Bernstein Diffusions for a Class of Linear Parabolic Partial Differential Equations

Pierre A. Vuillermot () and Jean C. Zambrini
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Pierre A. Vuillermot: Institut Élie Cartan
Jean C. Zambrini: Grupo de Física Matemática da Universidade de Lisboa

Journal of Theoretical Probability, 2014, vol. 27, issue 2, 449-492

Abstract: Abstract 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.

Keywords: Diffusion processes; Parabolic partial differential equations; 35K20; 60H30; 60K99 (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10959-012-0426-3

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