Techniques of quantum stochastic calculus are employed to factorise canonical Brownian motion on SO(3) into two non-commutative unitary operator valued processes acting in a suitable Fock space.
We replace the independent one-dimensional Brownian motions in the de nition of classical Levy area by the mutually noncommuting momentum and position Brownian motions of non-Fock quantum stochastic calculus, which are independent in the sense of factorisation of joint characteristic functions. The corresponding quantum Levy area can then be constructed in a way similar to Levys original construction as a quantum martingale limit. Acknowledgement 1 The author thanks the referee for drawing his attention to a number of mistypes. 1 Introduction. Levys stochastic area [10], [11], as well as being of great intrinsic interest and possessing numerous connections with classical mathematical analysis and mathematical physics [7], has recently come to prominence in the theory of rough paths [12], [13], [5]. Here we present a quantum or noncommutative version of Levy area in which the two independent one-dimensional Brownian motions used to de ne it are replaced by the pair of quantum Brownian motions (P;Q) ; forming the quantum Wiener process of [2]. Each of P and Q consists of a process of mutually commuting self-adjoint operators which can be simultaneously diagonalised as multiplication by a classical Brownian motion of appropriate variance in the Hilbert space of complex-valued functions square-integrable with respect to Wiener measure, But P and Q together satisfy the commutation relation P (s)Q(t) Q(t)P (s) = imin fs; tg I; (1)
Let P T be a stop time in symmetric Fock space over L Z (R + ).We show that to certain unitary operator valued processes U=(U(t),t^Q) which satisfy a stochastic differential equation driven by "non-commutative noise" in Fock space tensored with an "initial" Hilbert space, we can associate a stopped operator U(T).We use these operators to prove a "stopped cocycle relation" whereby for P T finite the process (U(T+t),t^Q) is factorised as the product of U(T) with a new process U T = (U T (t} , f^O) beginning "after time T".
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.