Functional ergodic limits for occupation time processes of site-dependent branching Brownian motions in ℝ

We consider a kind of site-dependent branching Brownian motions whose branching laws depend on the site-branching factor σ(·). We focus on the functional ergodic limits for the occupation time processes of the models in ℝ. It is proved that the limiting process has the form of λξ(·), where λ is the Lebesgue measure on ℝ and ξ(·) is a real-valued process which is non-degenerate if and only if σ is integrable. When ξ(·) is non-degenerate, it is strictly positive for t > 0. Moreover, ξ converges to 0 in finite-dimensional distributions if the integral of σ tends to infinity.