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

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  • Vuillermot, Pierre-A.
  • Zambrini, J.-C.

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 Schrödinger 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 several 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.

Suggested Citation

  • Vuillermot, Pierre-A. & Zambrini, J.-C., 2020. "On Bernstein processes generated by hierarchies of linear parabolic systems in Rd," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 2974-3004.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:5:p:2974-3004
    DOI: 10.1016/j.spa.2019.09.003
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    References listed on IDEAS

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    1. Alfred Galichon, 2016. "Optimal Transport Methods in Economics," Economics Books, Princeton University Press, edition 1, number 10870, December.
    2. Alfred Galichon, 2016. "Optimal transport methods in economics," Post-Print hal-03256830, HAL.
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