Operator-valued stochastic differential equations arising from unitary group representations (Q5937296)

Let \(\pi\) be a unitary representation of a Lie group \(G\) and \(\varphi (t)\) be a Lévy process in \(G\). Using analytic vector techniques it is shown that the unitary process \(U(t)=\pi(\varphi(t))\) 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 the operators of the form \(\pi(f)=\int f(g)\pi(g) dg\) 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.