On branching semi-groups, I
Nobuyuki Ikeda, Masao Nagasawa, Shinzo Watanabe · Proceedings of the Japan Academy Series A Mathematical Sciences · 1966
In the previous papers we have given a definition of branching Markov processes (abbreviated as B.M.P.) 1, discussed some fundamental equations of B.M.P. 2, and constructed B.M.P. in a probabilistic way 3, 4.This paper is a continuation of these papers and is devoted to an analytic construction of B.M.P.We shall treat this problem, however, in a little wider setup which may permit us to deal with the not necessarily positive branching semi- groups.(c.f.5).1. Definition of branching semi.groups.Let S be a compact Hausdorff space with countable base, S be the n-fold symmetric product of S (S-{}, an isolated point), and S= U S[J{z/} be the =0 one-point compactification of S'. ) We denote by C(S)(resp.C(S) =0 and C(S')) the space of bounded continuous functions on S (resp.on S and S').B(S) is the space of bounded Borel measurable functions on S. Co(S) (resp.Bo(S)) is the subspace of C(S) (resp.B(S)) the elements of which vanish at infinity z/.Definition 1.1.A contraction ) semi-group {T,; t_>_0} of linear