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

TL;DR

This paper investigates the long-term behavior of site-dependent branching Brownian motions in real space, revealing conditions for non-degenerate limits and characterizing their distributions through fractional integral equations.

Contribution

It establishes the ergodic limits for these processes and explicitly describes the limiting distributions using fractional integral equations and Levy measures.

Findings

Limiting processes are non-degenerate iff variance functions are integrable.

Limiting processes are positive, infinitely divisible, and self-similar.

Marginal distributions are characterized by 1/2-fractional integral equations.

Abstract

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.