Operator-Valued Stochastic Differential Equations Arising from Unitary Group Representations

Let π be a unitary representation of a Lie group G and (φ(t), t≥0) be a Lévy process in G. Using analytic vector techniques it is shown that the unitary process U(t)=π(φ(t)) satisfies an operator-valued stochastic differential equation. The prescription J(t) π(f)=U(t) π(f) U(t)* gives rise to an algebraic stochastic flow on the algebra generated by operators of the form π(f)=∫ f(g) π(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.