Mathematics Colloquia and Seminars

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A branching diffusion is a random process where trajectories follow, independently, a given diffusion law and a given fission law. On a $D$-dimensional hyperbolic (Lobachevsky) space, every individual trajectory of a homogeneous diffusion is attracted to a single (random) point of the absolute. However, if the fission rate is high (i.e., new branches are produced faster than they are attracted to the absolute), the whole random tree will withstand attraction (e.g., will visit every open set at indefinitely large times with probability one). The collection of limiting points for the random tree on the absolute can be characterised by its Hausdorff dimension (HD): under natural assumptions the HD of the limiting set equals, with probability one, a constant, and this constant can be calculated. The result is that for a low fission rate the HD increases from 0 to $(D-1)/2$ and then jumps to $D-1$, which exhibits a curious phase transition. The talk will discuss some recent results in this direction.