On invariant measures for simple branching processes (Summary)

Simple Branching Processes

Consider a particle that can replicate itself instantaneously at random times. In general there is a distribution of times from which the particle draws in order to initiate replication. For simplicity, we assume a Poisson process, and a replication rate of λ. Consider the total integer number of particles N(t) at any time t. Let us now consider one realization of the continuous-time branching ...

INVARIANT MEASURES FOR MARKOV PROCESSES(i)

Branching Processes, and Random-cluster Measures on Trees

Random-cluster measures on infinite regular trees are studied in conjunction with a general type of ‘boundary condition’, namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of randomcluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is enco...

Multifractal spectra for random self-similar measures via branching processes

Abstract Start with a compact set K ⊂ Rd. This has a random number of daughter sets each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to infinity, of th...

On Invariant Measures

Given a measurable transformation on a measure space one can ask whether or not there is an equivalent measure that is invariant under the transformation. This problem is discussed very thoroughly in Halmos' Lectures on ergodic theory, pp. 81-90, 97. The first result along these lines is due to E. Hopf who obtained necessary and sufficient conditions for the existence of a finite invariant meas...