On Bernstein processes generated by hierarchies of linear parabolic systems in Rd

TL;DR

This paper studies Bernstein processes generated by hierarchies of linear parabolic PDEs in Rd, exploring their properties, spectral measures, and connections to quantum mechanics, including the quantum harmonic oscillator.

Contribution

It introduces a framework for Bernstein processes from infinite hierarchies of linear parabolic systems and links them to spectral theory and quantum statistical mechanics.

Findings

Bernstein processes can be non-Markovian and stationary or non-stationary.

Spectral measures are associated with the pure point spectrum of elliptic operators.

Processes related to quantum harmonic oscillators are identified with Ornstein-Uhlenbeck processes.

Abstract

In this article we investigate the properties of Bernstein processes generated by infinite hierarchies of forward-backward systems of decoupled linear deterministic parabolic partial differential equations defined in Rd, where d is arbitrary. An important feature of those systems is that the elliptic part of the parabolic operators may be realized as an unbounded Schr\"odinger operator with compact resolvent in standard L2-space. The Bernstein processes we are interested in are in general non-Markovian, may be stationary or non-stationary and are generated by weighted averages of measures naturally associated with the pure point spectrum of the operator. We also introduce time-dependent trace-class operators which possess most of the attributes of density operators in Quantum Statistical Mechanics, and prove that the statistical averages of certain bounded self-adjoint observables usually evaluated by means of such operators coincide with the expectation values of suitable functions of the underlying processes. In the particular case where the given parabolic equations involve the Hamiltonian of an isotropic system of quantum harmonic oscillators, we show that one of the associated processes is identical in law with the periodic Ornstein-Uhlenbeck process.