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We study the long time behaviour of a large class of diffusion processes on$R^N$, generated by second order differential operators of (possibly)degenerate type. The operators that we consider {\em need not} satisfy theH\"ormander condition. Instead, they satisfy the so-called UFG condition,introduced by Herman, Lobry and Sussman in the context of geometric controltheory and later by Kusuoka and Stroock, this time with probabilisticmotivations. In this paper we study UFG diffusions and demonstrate theimportance of such a class of processes in several respects: roughly speakingi) we show that UFG processes constitute a family of SDEs which exhibitmultiple invariant measures and for which one is able to describe a systematicprocedure to determine the basin of attraction of each invariant measure(equilibrium state). ii) We use an explicit change of coordinates to prove thatevery UFG diffusion can be, at least locally, represented as a systemconsisting of an SDE coupled with an ODE, where the ODE evolves independentlyof the SDE part of the dynamics. iii) As a result, UFG diffusions areinherently "less smooth" than hypoelliptic SDEs; more precisely, we prove thatUFG processes do not admit a density with respect to Lebesgue measure on theentire space, but only on suitable time-evolving submanifolds, which wedescribe. iv) We show that our results and techniques, which we devised for UFGprocesses, can be applied to the study of the long-time behaviour ofnon-autonomous hypoelliptic SDEs and therefore produce several results on thislatter class of processes as well. v) Because processes that satisfy the(uniform) parabolic H\"ormander condition are UFG processes, our paper containsa wealth of results about the long time behaviour of (uniformly) hypoellipticprocesses which are non-ergodic, in the sense that they exhibit multipleinvariant measures.

Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes