SOME PROPERTIES OF QUANTUM LÉVY AREA IN FOCK AND NON-FOCK QUANTUM STOCHASTIC CALCULUS

Weconsider the analogueof Levy area, definedas an iterated stochastic integral, obtained by replacing two independent component one- dimensional Brownian motions by the mutually non-commuting momen- tum and position Brownian motions P and Q of either Fock or non-Fock quantum stochastic calculus, which are also stochastically independent in a certain sense. We show that the resulting quantum Levy area is trivially distributed in the Fock case, but has a non-trivial distribution in non-Fock quantum stochastic calculus which, after rescaling, interpolates between the trivial distribution and that of classical Levy area in the infinite tempera- ture limit. We also show that it behaves differently from the classical Levy area under a kind of time reversal, in both the Fock and non-Fock cases. 2000 AMS Mathematics Subject Classification: Primary: 81S25.