# On Bernstein processes of maximal entropy

## TL;DR

This paper defines and analyzes entropy-maximizing Bernstein stochastic processes linked to linear parabolic PDEs with Neumann boundary conditions, revealing Gibbs-type probability structures for maximal entropy processes.

## Contribution

It introduces a framework for statistical operators and entropy functionals for Bernstein processes related to linear PDE hierarchies, identifying Gibbs-type probabilities as maximal entropy solutions.

## Key findings

- Bernstein processes of maximal entropy have Gibbs-type probability sequences
- The framework applies to PDEs with self-adjoint Schrödinger operators and Neumann boundary conditions
- Illustrated with a hierarchy of heat equations in a 2D disk

## Abstract

In this article we define and investigate statistical operators and an entropy functional for Bernstein stochastic processes associated with hierarchies of forward-backward systems of decoupled deterministic linear parabolic partial differential equations. The systems under consideration are defined on open bounded domains $D\subset \mathbb{R}^{d}$ of Euclidean space where $d\in \mathbb{N}^{+}$ is arbitrary, and are subject to Neumann boundary conditions. We assume that the elliptic part of the parabolic operator in the equations is a self-adjoint Schr\"odinger operator,bounded from below and with compact resolvent in $L^{2}(D)$. The statistical operators we consider are then trace-class operators defined from sequences of probabilities associated with the point spectrum of the elliptic part in question, which allow the distinction between pure and mixed processes. We prove in particular that the Bernstein processes of maximal entropy are those for which the associated sequences of probabilities are of Gibbs type. We illustrate our results by considering processes associated with a specific hierarchy of forward-backward heat equations defined in a two-dimensional disk.