14:15
I will consider a stochastic process ( \xi_u; u \in
\Gamma_\infty ) where \Gamma_\infty is the set of vertices of an
infinite binary tree which satisfies some recursion relation
\xi_u= \phi(\xi_{u0},\xi_{u1}, \epsilon_u) \text { for each } u \in \Gamma_\infty.
Here u0 and u1 denote the two immediate daughters of the vertex u.
The random variables ( \epsilon_u; u\in \Gamma_\infty), which
are to be thought of as innovations, are supposed independent and
identically distributed. This type of structure is ubiquitous in models
coming from applied proability. A recent paper of Aldous and Bandyopadhyay
has drawn attention to the issue of endogeny: that is whether the process
( \xi_u; u \in \Gamma_\infty) is measurable with respect to the
innovations process. I will explain how this question is related to the
existence of certain dynamics and use this idea to develop a necessary and
sufficient condition [ at least if S is finite!] for endogeny in terms of
the coupling rate for a Markov chain on S^2 for which the diagonal is
absorbing.