seminar: Decomposition of diffusions
speaker: Kenneth David Elworthy (University of Warwick)abstract: This is taken from joint work with Yves LeJan and Xue-Mei Li on the general topic of "Geometric Filtering for Diffusions". The basic set up there is to have a smooth surjection of smooth manifolds each equipped with smooth diffusion generators, i.e. second order semi-elliptic differential operators with no zero order term. These are assumed to be intertwined by the surjection. In other words we are interested in filtering for diffusions when the observations form a diffusion. We also consider some filtering questions relating to diffusions on vector bundles in which case the generator will act on sections of a vector bundle e.g on differential forms.
The simplest non-trivial example is probably the case of Euclidean space with the origin removed taking the surjection to be just the Euclidean norm mapping onto the positive reals, with the generator on the domain space being one half times the Lapacian, and that on the target the generator of the induced Bessel process. However to tie in with random dynamical systems I shall concentrate on some examples coming from a stochastic flow on the target manifold. The results then give formulae for conditional expectations of functionals of the flow given the one-point motion from a given point. To do this it is convenient to consider the flow as an infinite dimensional diffusion taking values in a suitable diffeomorphism group.
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