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Free Brownian motion, free stochastic calculus and random matrices. (English) Zbl 0873.60056

Voiculescu, Dan-Virgil (ed.), Free probability theory. Papers from a workshop on random matrices and operator algebra free products, Toronto, Canada, Mars 1995. Providence, RI: American Mathematical Society. Fields Inst. Commun. 12, 1-19 (1997).
Summary: The free additive (resp. multiplicative) Brownian motion is a non-commutative stochastic process with free, stationary increments, which are distributed according to the semi-circular distributions (resp. the multiplicative analogues of semi-circular distributions). We prove that the free Brownian motions can be approximated by matrix valued Brownian motions. We also use stochastic calculus on a free Fock space to describe martingales associated with free additive Brownian motion, and we construct the free multiplicative Brownian motions as the stochastic exponential of free additive Brownian motion.
For the entire collection see [Zbl 0859.00025].

MSC:

60J65 Brownian motion
81S25 Quantum stochastic calculus