ON QUANTUM EXTENSIONS OF SEMIGROUPS OF BROWNIAN-MOTION ON A HALF-LINE

We study the problem of extending semigroups of classical Brownian mot ion on a half-line with various boundary conditions to a quantum dynam ical semigroup on the algebra of all bounded operators on L(2)(0, +inf inity). We find necessary conditions for the infinitesimal generator t o be representable in the Davies-Lindblad form and construct an extens ion of the semigroup of classical Brownian motion with reflection at t he point 0.