  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.
