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Journal of Applied Analysis

A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges

David Applebaum

Published Jan 1, 2007• 1 citation
Publisher

Abstract

We introduce “probabilistic” and “stochastic Hilbertian structures”. These seem to be a suitable context for developing a theory of “quantum Gaussian processes”. The Schauder system is utilised to give a Lévy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.

DOI
10.1515/jaa.2007.275

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We introduce “probabilistic” and “stochastic Hilbertian structures”. These seem to be a suitable context for developing a theory of “quantum Gaussian processes”. The Schauder system is utilised to give a Lévy-Ciesielski representation of quantum (bosonic) Brownian motion as operators in Fock space over a space of square summable sequences. Similar results hold for non-Fock, fermion, free and monotone Brownian motions. Quantum Brownian bridges are defined and a number of representations of these are given.
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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.