Spectral Representation for Branching Processes on the Real Half Line

Abstract

The spectral theory for the semigroups of Galton-Watson processes (G.W.P.'s) was developed by Karlin and McGregor [4] [5]. On the other hand, a continuous state branching process (C.B.P.), which may be regarded as a continuous version of a G.W.P., was introduced by Jirina [3] and recently discussed by Lamperti [7], S. Watanabe [8], etc. The object of this paper is to obtain the spectral representation theorems for C.B.P.'s similar to those of Karlin and McGregor for G.W.P.'s. These may be of some interest, since there are many C.B.P.'s with discontinuous sample functions and their semigroups are nonsymmetrizable. In § 1, we obtain a representation (see (1.10)) of a so called ^-semigroup by means of the stationary measure, under the assumption that the extinction probability is positive. In § 2, we shall prepare several lemmas. The most important one is Lemma 2.4, which asserts that, under Condition A, the representation of a M>semigroup in § 1 turns out to be a spectral representation of the transition function. In §3, the spectral representation theorem for sub- and supercritical C.B.P.'s is given. In this representation, only the discrete spectrum appears which consists of powers of the largest eigenvalue. In § 4, the critical case is considered. In this case, the spectrum is continuous. The asymptotic behavior of the semigroup is also obtained. In § 5, we give a few examples which contain all diffusion C.B.P.'s. The author wishes to express his thanks to Professor K. Ito