Fock space decomposition of Levy processes

by R. F. Streater. Mini-Proceedings of 2nd MaPhySto conference on Levy Processes, Aarhus University, Jan 2002. Levy Processes, Theory and Applications, Ed Lars Madsen, pp 242-246, Miscellanea no. 22, 2002. It can be read in the archive as math.PR/0307243.

ABSTRACT

We apply the theory of infinitely divisible representations of groups to the case when the group is R, the additive group of real numbers. By the general result, the Hilbert space can be embedded in a Fock space. We show that the classification in terms of cocycles leads to the Levy-Khinchin formula. De Finetti's formula just gives the coboundaries, while Kolmogorov's formula gives those cocycles f that obey the extra condition that xf(x) is square integrable. This condition allows some group cocycles that are not coboundaries, arising when f is not square-integrable at x=0; it is exactly the condition required in an earlier paper for f to be a cocycle in the sense of Lie algebras. Kolmogorov's work is thus a classification of the infinitely divisible cyclic representations of the Lie algebra R. The remaining group cocycles, that are not algebra-cocycles, arise when xf(x) is square-integrable at x=0 but not at infinity. These are included in Levy's classification.