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 $\partialH^d$. Under these assumptions, a formula is established for the Hausdorff dimension $h(\Lambda)$ of the limiting (random) set $\Lambda\subseteq\partialH^d$, 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 $H^d$.