Entangling and disentangling noise and dynamics The question of underlying geometric structures

Modeling noise in filtering problems is often done by means of stochastic differential equations (sde) driven by Brownian motion. The signal is thus a Markov process and, in particular, a diffusion process under general regularity assumptions. However, practical engineering problems requiring filtering often start as deterministic systems. Hence the problem of “imbedding” a deterministic system – typically a set of differential equations (coming from mechanics, etc.) – into a sde or a diffusion process is set. We first examine current practices in the field and questions they raise. For instance, we discuss the approach consisting in adding a Brownian motion to an ordinary differential equation to get a sde. Following this, we briefly present the theory of sdes on manifolds as developped by Emery. Then, we recall different physical and mathematical sources of diffusion processes. Mathematically, these latter appear as limits of deterministic systems (deterministic interacting particles systems) or of stochastic systems (randomly perturbed differential systems). Physically, diffusion may be a surface term in a conservation equation. By focusing on the sources of diffusion processes, we take the opportunity to enlighten the role played by implicit underlying differential geometric structures (Riemannian metrics). We then show different ways in which noise and dynamics can be entangled and disentangled. We suggest that connexions or Riemannian metrics are more or less inevitable geometric objects needed to separate “drift” and “noise” in a diffusion process. We end up by presenting different applications. Riemannian geometric tools are used to analyze Zakai equation of filtering: the existence of symmetries reveals certain physical and geometric properties not always explicit; now, such symmetries may simplify the resolution of Zakai equation which may have finite dimensional filters as solutions ([Coh97b, Coh97a]), or group invariant solutions ([Coh98]). We also make use ∗6 et 8 avenue Blaise Pascal, Cité Descartes, Champs sur Marne — 77455 Marne La Vallée Cedex 2, France. [email protected]