Functional ergodic theorems of site-dependent branching Brownian motions in R

Details Functional ergodic theorems of site-dependent branching Brownian motions in R Functional ergodic theorems of site-dependent branching Brownian motions in R

2011-07-26

arxiv.org/abs/1107.5190v2

Mathematics

Probability

25 pages

Scientific paper

In this paper, we studied the functional ergodic limits of the site-dependent branching Brownian motions in R. The results show that the limiting processes are non-degenerate if and only if the variance functions of branching laws are integrable. When the functions are integrable, although the limiting processes will vary according to the integrals, they are always positive, infinitely divisible and self-similar, and their marginal distributions are determined by a kind of 1/2-fractional integral equations. As a byproduct, the unique non-negative solutions of the integral equations are explicitly presented by the Levy-measure of the corresponding limiting processes.

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