The Geometry of Filtering

TL;DR

This paper explores the geometric structures formed by diffusion operators on different spaces connected by smooth maps, with applications to stochastic flows, filtering, and principal bundles, revealing new insights into their decompositions and curvatures.

Contribution

It introduces a unified geometric framework for analyzing diffusion operators interconnected by smooth maps, including non-linear semi-connections and decompositions relevant to filtering and stochastic flows.

Findings

Canonical decomposition of operators on domain spaces

Generalized Wietzenbock curvatures in equivariant cases

Skew-product decompositions of diffusion processes

Abstract

Geometry arising from two diffusion operators (smooth semi-elliptic, second order differential operators) on different spaces but intertwined by a smooth map is described. Particular cases arise from Riemannian submersions when the operators are Laplace-Beltrami operators, from equivariant operators on the total space of a principal bundle, and for the operators on the diffeomorphism group arising from stochastic flows. Classical non-linear filtering problems also lead to such conffigurations. A basic tool is the, possibly, non-linear "semi-connection" induced by this set up, leading to a canonical decomposition of the operator on the domain space. Topics discussed include: generalised Wietzenbock curvatures arising in the equivariant case, skew -product decompositions of diffusion processes, conditioned processes, classical filtering, decomposition of stochastic flows, and connections determined by stochastic differential equations.