A Fermion Lévy theorem
Lévy's theorem in classical probability theory asserts that any martingale \((M_ t)\), with \(M_ 0=0\), continuous sample paths and such that \(M^ 2_ t-t\) is a martingale, has the same law as standard Brownian motion from 0. The problem is to extend such a theorem to non- commutative probability. One of the main difficulties is to translate the path continuity condition, which the authors interpret as the identity of two natural quadratic variation processes, a ``square bracket'' and an ``angle bracket''. Using this idea, the authors are able to prove a Lévy theorem characterizing Fermion Brownian motion, under an auxiliary hypothesis on the existence of fourth order moments.
Lévy theorem characterizing Fermion Brownian motion
scientific article; zbMATH DE number 220680
scientific article