On some Gaussian Bernstein processes in RN and the periodic Ornstein-Uhlenbeck process

TL;DR

This paper establishes the existence of Gaussian Bernstein processes linked to parabolic equations in Euclidean space, focusing on quantum harmonic oscillators and introducing new non-Markovian processes related to the periodic Ornstein-Uhlenbeck process.

Contribution

It provides new existence results for Bernstein processes associated with quantum harmonic oscillator Hamiltonians, including Gaussian, non-Markovian, and periodic Ornstein-Uhlenbeck processes, expanding the theoretical framework.

Findings

Existence of Gaussian Bernstein processes for quantum harmonic oscillator systems.

Introduction of a new class of stationary non-Markovian processes.

Connection of these processes to statistical mechanics and pattern theory.

Abstract

In this article we prove new results regarding the existence of Bernstein processes associated with the Cauchy problem of certain forward-backward systems of decoupled linear deterministic parabolic equations defined in Euclidean space of arbitrary dimension N, whose initial and final conditions are positive measures. We concentrate primarily on the case where the elliptic part of the parabolic operator is related to the Hamiltonian of an isotropic system of quantum harmonic oscillators. In this situation there are many Gaussian processes of interest whose existence follows from our analysis, including N-dimensional stationary and non-stationary Ornstein-Uhlenbeck processes, as well as a Bernstein bridge which may be interpreted as a Markovian loop in a particular case. We also introduce a new class of stationary non-Markovian processes which we eventually relate to the N-dimensional periodic Ornstein-Uhlenbeck process, and which is generated by a one-parameter family of non-Markovian probability measures. In this case our construction requires an infinite hierarchy of pairs of forward-backward heat equations associated with the pure point spectrum of the elliptic part, rather than just one pair in the Markovian case. We finally stress the potential relevance of these new processes to statistical mechanics, the random evolution of loops and general pattern theory.