Intertwinned Diffusions by Examples.

We discuss the geometry induced by pairs of diffusion operators on two states spaces related by a map from one space to the other. This geometry led us to an intrinsic point of view on filtering. This will be explained plainly by examples, in local coordinates and in the metric setting. This article draws largely from the books (˵Elworthy et al., On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics 1720, Springer, 1999″, ˵Elworthy et al., The Geometry of Filtering. To appear in Frontiers in Mathematics Series, Birkhauser″) and aims to have a comprehensive account of the geometry for a general audience. [ABSTRACT FROM AUTHOR]