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Birkhäuser Basel eBooks

Ornstein-Uhlenbeck Processes as Bernstein Processes

Ana Bela Cruzeiro, Jean-Claude Zambrini

Published Jan 1, 1993• 6 citations
Publisher

Abstract

Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-0348-8555-3_3

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Bernstein diffusions belong to a new class of time symmetric (but not time homogeneous) stochastic processes associated with the quantum dynamics of nonrelativistic particles in potentials. It is shown in which sense Feynman’s formal path integral method can be interpreted in terms of those processes, specially for the subset of Gaussian Bernstein diffusions. The familiar Ornstein-Uhlenbeck process becomes, in this framework, a particular Gaussian process in a large class of Bernstein diffusions, all associated with the same dynamics. The method is also illustrated for some Hilbert space valued Gaussian Diffusions These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
A new class of Gaussian (‘‘Bernstein’’) diffusion processes associated with semiclassical quantum dynamics is introduced and studied according to a new Euclidean strategy for the quantization of systems of particles in a potential (‘‘Euclidean quantum mechanics’’). All the measures known by other means, up to now, are included as particular cases of the present framework. It is shown that the theory may be regarded as a well-defined probabilistic version of Feynman’s space-time approach distinct, in particular, from the one due to Kac, and with much closer relations with classical Lagrangian dynamics. Many explicit examples are considered.
A general description of Bernstein processes, a class of diffusion processes, relevant to the probabilistic counterpart of quantum theory known as Euclidean Quantum Mechanics, is given. It is compatible with finite or infinite dimensional state spaces and singular interactions. Although the relations with statistical physics concepts (Gibbs measure, entropy,…) is stressed here, recent developments requiring Feynman’s quantum mechanical tools (action functional, path integrals, Noether’s Theorem,…) are also mentioned and suggest new research directions, especially in the geometrical structure of our approach.
In Section 21 we have already seen how the classical stochastic processes with independent increments can be realised as suitable linear combinations of the creation, conservation and annihilation operators in the boson Fock space Γ s (ℋ) over a Hilbert space ℋ. This includes, in particular, the Brownian motion and Poisson process. Since a well-developed theory of stochastic integration with respect to these classical processes exists, it is natural to wonder whether it would be possible to introduce quantum probabilistic analogues of stochastic integrals with respect to processes defined by the creation, conservation and annihilation operators. To this end one needs the notions of filtration and adapted processes. This, in turn, depends on the notion of time. We adopt the view that time is an ℝ+-valued observable in a Hilbert space ℋ which, as a spectral measure, has no jumps. The absence of jumps is assumed more as a matter of mathematical convenience than a philosophical or conceptual necessity. We shall now examine how such a notion of time leads to a filtration and the definition of adapted processes. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.