Stationary Measures for Branching Processes

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 large deviations of empirical measures for stationary Gaussian processes

We show that the Large Deviation Principle with respect to the weak topology holds for the empirical measure of any stationary continuous time Gaussian process with continuous vanishing at in nity spectral density. We also point out that Large Deviation Principle might fail in both continuous and discrete time if the spectral density is discontinuous.

The Existence Op Stationary Measures for Certain Markov Processes

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...

Smaller population size at the MRCA time for stationary branching processes

Abstract. We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck ...