Filtered probability (Q1600616)

In this paper, the author presents his theory of filtered random variables, which provides in particular an interpolation between non-commutative probabilistic settings such as Boolean and free probability. Given a family of noncommutative probability spaces of the form \((\widehat{\mathcal A} , \widehat{\varphi}) = (\bigotimes_{l\in L} {\mathcal A}_l , \bigotimes_{l\in L} \varphi_l)\), he constructs for each \(l\in L\) an enlargement \((\widetilde{\mathcal A}_l,\widetilde{\varphi}_l)\) of \(({\mathcal A}_l,\varphi_l)\), and formulates a notion of independence on \((\bigotimes_{l\in L} \widetilde{\mathcal A}_l , \bigotimes_{l\in L} \widetilde{\varphi}_l)\), depending on the choice of a family \((P_l)_{l\in L}\) of projectors. Boolean and free independent random variables can be constructed as finite or infinite sums of filtered random variables which are elements of \(\bigotimes_{l\in L} \widetilde{\mathcal A}_l\). The analogs of the fundamental quantum stochastic processes, stochastic calculus, and limit theorems, are formulated in this framework.