David G. Kendall

On the use of the characteristic functional in the analysis of some stochastic processes occurring in physics and biology

1. Introduction. In a recent contribution to these Proceedings Alladi Rama-krishnan (23) has discussed a number of problems which arise when the development of a cascade shower of cosmic rays is considered from the standpoint of the theory of stochastic processes. As has often been remarked (see, for example, the important study by Niels Arley (1)), there is a close formal analogy between such physical phenomena and the growth of biological populations. In particular, if the distance of penetration ( t ) is identified with the time, and the energy ( E ) of a particle in the shower is replaced by the age ( x ) of an individual in the population, there emerges an obvious analogy between stochastic fluctuations in the energy spectrum of the shower and similar fluctuations in the age distribution of the population.

Some Ergodic Theorems for One-Parameter Semigroups of Operators

If ^ = {7^: t ^0} is a one-parameter semigroup of operators on a Banach space X, an element x of X is called ergodic if TtX has a generalized limit as t -> oo. It is shown, for a wide class of semigroups, that the use of Abel or Cesaro limits, and of weak or strong convergence, leads to four equivalent definitions of ergodicity. When the resolvent operator of G has suitable compactness properties, every element of X is ergodic. The ergodic properties of G can be completely determined when its infinitesimal generator is known. Some of these results can be extended to more generaltypes of weak convergence in X, and this leads to a discussion of ergodic properties of the semigroup adjoint to G