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In this article dedicated to the memory of Igor D. Chueshov, I firstsummarize in a few words the joint results that we obtained over a period ofsix years regarding the long-time behavior of solutions to a class ofsemilinear stochastic parabolic partial differential equations. Then, as thebeautiful interplay between partial differential equations and probabilitytheory always was close to Igor's heart, I present some new results concerningthe time evolution of certain Markovian Bernstein processes naturallyassociated with a class of deterministic linear parabolic partial differentialequations. Particular instances of such processes are certain conditionedOrnstein-Uhlenbeck processes, generalizations of Bernstein bridges andBernstein loops, whose laws may evolve in space in a non trivial way.Specifically, I examine in detail the time development of the probability offinding such processes within two-dimensional geometric shapes exhibitingspherical symmetry. I also define a Faedo-Galerkin scheme whose ultimate goalis to allow approximate computations with controlled error terms of the variousprobability distributions involved.

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