Markov branching processes and semigroups of operators

AbstractIn this paper we study the semigroups of operators associated with Markov branching processes. Our approach is based on the semigroup of operators associated with the generating function of the probabilities of a given branching process. Let F(s, t) = ∑x = 0∞ Px(t)sx, ¦ s ¦ ⩽ 1, denote the generating function of the probabilities {Px(t)}. We consider F(s, t), for every fixed t, to be an element of the Hilbert space H2 of functions analytic in the unit circle, and define an endomorphism T(t) on H2 as follows: For F(s, t) ϵ H2, T(t)[F(s, u)] = F(F(s, t), u) = F(s, t + u). If F(s, 0) = s, then the above becomes F(s, t) = T(t)[s], It is shown that the family of endomorphisms {T(t), t ⩾ 0} is a semigroup of endomorphisms in H2 and various properties of the semigroup and its infinitesimal generator are established