Titles and Abstracts Stability of the Nonlinear Filter for Random Expanding Maps

In the first part of the talk, I introduce sharp gradient bounds for the perturbed diffusion semigroup. In contrast with existing results, the perturbation studied here is random and the bounds obtained are pathwise. The approach builds on the classical work of Kusuoka and Stroock. It extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalized conditional distribution of a partially observed signal. by a random process. It uses a pathwise representation of the perturbed semigroup in the spirit of the one introduced by Ocone. The estimates we derive have sharp short time asymptotics. In the second part of the talk, I introduce a class of particle approximations (particle filters) to the (normalized) perturbed diffusion semigroup. The method combines the Kusuoka–Lyons–Victoir (KLV) cubature method on Wiener space to approximate the law of the signal with a minimal variance ‘thinning’ method, called the tree-based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of the approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle filter in terms of the computational effort (number of particles) and the time discretization grid mesh. This is joint work with C. Litterer (Imperial), S. Ortiz-Latorre (Oslo) and T. Lyons (Oxford). Two non-self-adjoint specral problems