Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes
File(s)
Published version
Author(s)
Cass, Thomas
Crisan, Dan
Dobson, Paul
Ottobre, Michela
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.
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.
Date Issued
2021-03-23
Date Acceptance
2020-12-23
Citation
Electronic Journal of Probability, 2021, 26, pp.1-72
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/).
License URL
Sponsor
Engineering & Physical Science Research Council (E
Grant Number
RF060148 - EP/K040154/1
Subjects
Science & Technology
Physical Sciences
Statistics & Probability
Mathematics
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
VARIANCE REDUCTION
MALLIAVIN CALCULUS
EQUATION
BOUNDS
math.PR
math.PR
Statistics & Probability
0104 Statistics
0105 Mathematical Physics
Publication Status
Published
Article Number
22
Date Publish Online
2021-03-23