Pierre-A. Vuillermot 1,2,
Inst. Élie Cartan de Lorraine, UMR-CNRS 7502, Nancy, France
Dept. de Matemática, Universidade de Lisboa, Lisboa, Portugal
Received January 2017 Revised May 2017 Published May 2018 Early access February 2018
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.
Citation: Pierre-A. Vuillermot. On the time evolution of Bernstein processes associated with a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1073-1090. doi: 10.3934/dcdsb.2018142
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Pierre-A. Vuillermot