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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.

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