Summary: A L´evy-Cielsielski Expansion for Quantum
Brownian Motion and the Construction of
Quantum Brownian Bridges
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
Dedicated to Robin Hudson on his 65th birthday.
We introduce "probabilistic" and "stochastic Hilbertian structures".
These seem to be a suitable context for developing a theory of "quan-
tum Gaussian processes". The Schauder system is utilised to give a
L´evy-Cielsielski representation of quantum Brownian motion as oper-
ators in Fock space over a space of square summable sequences. Quan-
tum Brownian bridges are defined and a number of representations of
these are given.
Keywords and phrases: daggered space, probabilistic Hilbertian
structure, stochastic Hilbertian structure, Fock space, exponential