FEYNMAN-KAC REPRESENTATION OF SOME NONCOMMUTATIVE ELLIPTIC-OPERATORS

Gaussian averages of automorphisms of a von Neumannn algebra yield Mar kov semigroups by the well-known procedure of subordination. We constr uct operator-valued martingales to realise perturbations of such semig roups through Feynman-Kac formulae. The perturbations are noncommutati ve vector fields, and the martingales are operator families, which are determined by an Ito equation on each vector and satisfy cocycle rela tions with respect to a randomised flow on the algebra. In particular this gives a probabilistic representation of some symmetric Markov sem igroups considered by Davies and Lindsay. (C) 1997 Academic Press.