Feynman–Kac Representation of Some Noncommutative Elliptic Operators

Gaussian averages of automorphisms of a von Neumannn algebra yield Markov semigroups by the well-known procedure of subordination. We construct operator-valued martingales to realise perturbations of such semigroups through Feynman–Kac formulae. The perturbations are noncommutative vector fields, and the martingales are operator families, which are determined by an Itô equation on each vector and satisfy cocycle relations with respect to a randomised flow on the algebra. In particular this gives a probabilistic representation of some symmetric Markov semigroups considered by Davies and Lindsay.