Survey on a quantum stochastic extension of Stone's Theorem

Abstract

From K\"ummerer's investigations on stationary Markov processes has emerged an operator algebraic definition of white noises which captures many examples from classical as well as from non-commutative probability. Within non-commutative $L^p$-spaces associated to a white noise, the role of (non-)commutative L\'evy processes is played by additive cocycles for the white noise shift, and moreover, the notion for exponentials of classical L\'evy processes is generalized by unitary cocycles. As a main result we report a bijective correspondence between additive and unitary cocycles for white noise shifts. If the cocycles are required to be differentiable, the presented correspondence reduces to Stone's theorem (for norm continuous unitary groups). The correspondence needs the development of background results for additive cocycles with $L^\infty$-bounded covariance operators: an operator-valued stochastic It\^o integration, quadratic variations and non-commutative martingale inequalities as well as stochastic differentiation. Related results and recent progress towards the case of additive cocycles with unbounded variance operators are reported.