On the time evolution of Bernstein processes associated with a class of parabolic equations

Pierre-A. Vuillermot

doi:10.3934/dcdsb.2018142

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2018, Volume 23, Issue 3: 1073-1090. Doi: 10.3934/dcdsb.2018142

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On the time evolution of Bernstein processes associated with a class of parabolic equations

Pierre-A. Vuillermot^1,2,

1.
Inst. Élie Cartan de Lorraine, UMR-CNRS 7502, Nancy, France
2.
Dept. de Matemática, Universidade de Lisboa, Lisboa, Portugal

Received: January 2017

Revised: May 2017

Early access: February 2018

Published: July 2018

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Abstract

In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic parabolic partial differential equations. Then, as the beautiful interplay between partial differential equations and probability theory always was close to Igor's heart, I present some new results concerning the time evolution of certain Markovian Bernstein processes naturally associated with a class of deterministic linear parabolic partial differential equations. Particular instances of such processes are certain conditioned Ornstein-Uhlenbeck processes, generalizations of Bernstein bridges and Bernstein loops, whose laws may evolve in space in a non trivial way. Specifically, I examine in detail the time development of the probability of finding such processes within two-dimensional geometric shapes exhibiting spherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goal is to allow approximate computations with controlled error terms of the various probability distributions involved.

Keywords:

Bernstein processes,
parabolic partial differential equations.

Mathematics Subject Classification: 35K15, 60G15, 65D15.

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