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

## TL;DR

This paper explores the time evolution of Bernstein processes linked to linear parabolic PDEs, including conditioned Ornstein-Uhlenbeck processes, with a focus on their spatial probability distribution changes and a proposed approximation scheme.

## Contribution

It introduces new results on the dynamics of Bernstein processes associated with linear parabolic PDEs and develops a Faedo-Galerkin scheme for their probabilistic computations.

## Key findings

- Analysis of probability evolution within spherical regions
- Development of a Faedo-Galerkin approximation method
- Insights into the spatial behavior of Bernstein processes

## 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.