Journal of Applied Analysis 13 (2007), No. 2, 275--290
Copyright Heldermann Verlag 2007
A Lévy-Ciesielski Expansion for Quantum Brownian Motion and the Construction of Quantum Brownian Bridges
David B. Applebaum
Probability and Statistics Dept., University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, England
d.applebaum@sheffield.ac.uk
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.
Keywords: Daggered space, probabilistic Hilbertian structure, stochastic Hilbertian structure, Fock space, exponential vector, quantum Brownian motion, Haar system, Schauder system, Levy-Ciesielski expansion, quantum Brownian bridge.
MSC: 81S25, 60H99, 42C40, 81S05
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