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Feynman–Kac Representation of Some Noncommutative Elliptic Operators

✍ Scribed by J.Martin Lindsay; Kalyan B. Sinha

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
375 KB
Volume
147
Category
Article
ISSN
0022-1236

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✦ Synopsis

Gaussian averages of automorphisms of a von Neumannn algebra yield Markov semigroups by the well-known procedure of subordination. We construct operatorvalued 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 Ito^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.

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