Large-Time Behavior of a Branching Diffusion on a Hyperbolic Space

In this paper we consider a general hyperbolic branching diffusion on a Lobachevsky space $H^d$. The question is to evaluate the Hausdorff dimension of the limiting set on the boundary (i.e., absolute) $\partialH^d$. In the case of a homogeneous branching diffusion, an elegant formula for the Hausdorff dimension was obtained by Lalley and Sellke [Probab. Theory Related Fields, 108 (1997), pp. 171–192] for $d=2$ and by Karpelevich, Pechersky, and Suhov [Commun. Math. Phys., 195 (1998), pp. 627–642] for a general $d$. Later on, Kelbert and Suhov [Probab. Theory Appl., 51 (2007), pp. 155–167] extended the formula to the case where the branching diffusion was in a sense asymptotically homogeneous (i.e., its main relevant parameter, the fission potential, approached a constant limiting value near the absolute). In this paper we show that the Hausdorff dimension of the limiting set can be bounded from above and below in terms of the maximum and minimum points of the fission potential. As in [M. Kelbert and Y. M. Suhov, Probab. Theory Appl., 51 (2007), pp. 155–167], the method is based on properties of the minimal solution to a Sturm–Liouville problem with general potential and elements of the harmonic analysis on $H^d$. We also relate the Hausdorff dimension with properties of recurrence and transience of a branching diffusion, as was defined by Grigoryan and Kelbert [Ann. Probab., 31 (2003), pp. 244–284] on a general-type manifold.