Operator-Valued Stochastic Differential Equations Arising from Unitary Group Representations

Let (pi)be a unitary representation of a Lie group G and ((null set)(t), t>=0) be a Levy process in G. Using analytic vector techniques it is shown that the unitary process U(t)=(pi)((null set)(t)) satisfies an operator-valued stochastic differential equation. The prescription J(t) (pi)(f)=U(t) (pi)(f) U(t)* gives rise to an algebraic stochastic flow on the algebra generated by operators of the form (pi)(f)=(integral) f(g) (pi)(g) dg where f is in the group algebra and dg is a left Haar measure. J(t) itself satisfies an operator-valued stochastic differential equation of a type which has been previously studied within the context of quantum stochastic calculus.