Branching diffusions on H^d with variable fission: The hausdorff dimension of the limiting set

This paper extends results of previous papers [S. Lalley and T. Sellke, Probab. Theory Related Fields, 108 (1997), pp. 171-192] and [F. I. Karpelevich, E. A. Pechersky, and Yu. M. Suhov, Comm. Math. Phys., 195 (1998), pp. 627-642] on the Hausdorff dimension of the limiting set of a homogeneous hyperbolic branching diffusion to the case of a variable fission mechanism. More precisely, we consider a nonhomogeneous branching diffusion on a Lobachevsky space H ^d and assume that parameters of the process uniformly approach their limiting values at the absolute ∂H^d. Under these assumptions, a formula is established for the Hausdorff dimension h(Λ) of the limiting (random) set Λ ⊆ ∂H^d, which agrees with formulas obtained in the papers cited above for the homogeneous case. The method is based on properties of the minimal solution to a Sturm-Liouville equation, with a potential taking two values, and elements of the harmonic analysis on H ^d.