  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.
