  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.
