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

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Title: Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes
Authors: Cass, T
Crisan, D
Dobson, P
Ottobre, M
Item Type: Journal Article
Abstract: 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 need not satisfy the Hörmander Condition (HC). Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock. We demonstrate the importance of the class of UFG processes in several respects: i) we show that UFG processes constitute a family of SDEs which exhibit, in general, multiple invariant measures (i.e. they are in general non-ergodic) and for which one is able to describe a systematic procedure to study the basin of attraction of each invariant measure (equilibrium state). ii) We use an explicit change of coordinates to prove that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently “less smooth” than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. iv) We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produce several results on this latter class of processes as well. v) Because processes that satisfy the (uniform) parabolic HC are UFG processes, this paper contains a wealth of results about the long time behaviour of (uniformly) hypoelliptic processes which are non-ergodic.
Issue Date: 23-Mar-2021
Date of Acceptance: 23-Dec-2020
URI: http://hdl.handle.net/10044/1/86419
DOI: 10.1214/20-EJP577
ISSN: 1083-6489
Publisher: Institute of Mathematical Statistics
Start Page: 1
End Page: 72
Journal / Book Title: Electronic Journal of Probability
Volume: 26
Copyright Statement: © 2021 the authors. Rights: Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/).
Sponsor/Funder: Engineering & Physical Science Research Council (E
Funder's Grant Number: RF060148 - EP/K040154/1
Keywords: Science & Technology
Physical Sciences
Statistics & Probability
diffusion semigroups
parabolic PDE
UFG condition
Hormander condition
long time asymptotics
processes with multiple invariant measures
non-ergodic SDEs
distributions with non-constant rank
stochastic control theory
Statistics & Probability
0104 Statistics
0105 Mathematical Physics
Publication Status: Published
Article Number: 22
Online Publication Date: 2021-03-23
Appears in Collections:Financial Mathematics
Pure Mathematics
Faculty of Natural Sciences

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