Chaoticexpansion of elements of the universal enveloping algebra of a lie algebra associated with a quantumstochastic calculus

Abstract

The functional Ito formula, in the form $df({\bf \Lambda})=f({\bf \Lambda} +d{\bf \Lambda})-f({\bf \Lambda} )$, is formulated and proved in the context of a Lie algebra $\mathcal{L}$ associated with a quantum (non-commutative) stochastic calculus. Here $f$ is an element of the universal enveloping algebra $\mathcal{U}$ of $\mathcal{L}$, and $f({\bf \Lambda} + d{\bf \Lambda})-f({\bf \Lambda} )$ is given a meaning using the coproduct structure of $\mathcal{U}$ even though the individual terms of this expression have no meaning. The Ito formula is equivalent to a chaotic expansion formula for $f({\bf \Lambda} )$ which is found explicitly.

1991 Mathematics Subject Classification: primary 81S25; secondary 60H05; tertiary 18B25.