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Hausdorff dimension of the limiting set for a branching diffusion on the hyperbolic space
Probability| Speaker: | Yuri Suhov |
| Location: | 693 Kerr |
| Start time: | Tue, Oct 4 2005, 3:10PM |
Description
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.