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Large-time behavior of a branching diffusion on a hyperbolic space. (English. Russian original) Zbl 1161.60332
Theory Probab. Appl. 52, No. 4, 594-613 (2008); translation from Teor. Veroyatn. Primen. 52, No. 4, 660-684 (2007).
Authors’ abstract: We consider a general hyperbolic branching diffusion on a Lobachevsky space \({\mathbf H}^d\). The question is to evaluate the Hausdorff dimension of the limiting set on the boundary (i.e., absolute) \(\partial{\mathbf H}^d\). In the case of a homogeneous branching diffusion, an elegant formula for the Hausdorff dimension was obtained by S. P. Lalley and T. Sellke [Probab. Theory Related Fields, 108, 171–192 (1997; Zbl 0883.60092)] for \(d=2\) and by F. I. Karpelevich, E. A. Pechersky, and Yu. M. Suhov [Commun. Math. Phys., 195 , 627–642(1998; Zbl 0933.60084)] for a general \(d\). Later on, M. Kelbert and Yu. M. Suhov [Probab. Theory Appl., 51, 155–167 (2007; Zbl 1112.60081)] 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 [Zbl 1112.60081], 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 \({\mathbf H}^d\). We also relate the Hausdorff dimension with properties of recurrence and transience of a branching diffusion, as was defined by A. Grigoryan and M. Kelbert [Ann. Probab., 31, 244–284 (2003; Zbl 1014.60081)] on a general-type manifold.

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 Applications of branching processes
58J65 Diffusion processes and stochastic analysis on manifolds
28A78 Hausdorff and packing measures
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