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

@article{Cass2021LongtimeBO,
  title={Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes},
  author={Thomas Cass and Dan Crisan and Paul Dobson and Michela Ottobre},
  journal={Electronic Journal of Probability},
  year={2021}
}
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 the Hormander condition. 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, this time with probabilistic motivations. In this paper we study UFG diffusions and… 

Figures from this paper

Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability

We study Poisson equations and averaging for Stochastic Differential Equations (SDEs). Poisson equations are essential tools in both probability theory and partial differential equations (PDEs).

Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups

We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires i) exponential decay in time of the

Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

TLDR
The objective of this paper is to control the distance of the standard Euler scheme with decreasing step to the invariant distribution of such an ergodic diffusion with possibly multiplicative diffusion term (non-constant diffusion coefficient).

A non-linear kinetic model of self-propelled particles with multiple equilibria

We introduce and analyse a continuum model for an interacting particle system of Vicsek type. The model is given by a non-linear kinetic partial differential equation (PDE) describing the

Efficient Bayesian computation for low-photon imaging problems

This paper studies a new and highly efficient Markov chain Monte Carlo (MCMC) methodology to perform Bayesian inference in low-photon imaging problems, with particular attention to situations

Fast Non-mean-field Networks: Uniform in Time Averaging

TLDR
It is proved that the evolution of the particles' empirical density is described (after taking both limits) by a non-linear Fokker-Planck equation; the heart of the proof consists of controlling precisely the dependence in $N$ of the averaging estimates.

References

SHOWING 1-10 OF 56 REFERENCES

Pointwise gradient bounds for degenerate semigroups (of UFG type)

  • D. CrisanM. Ottobre
  • Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2016
In this paper, we consider diffusion semigroups generated by second-order differential operators of degenerate type. The operators that we consider do not, in general, satisfy the Hörmander condition

Sharp gradient bounds for the diffusion semigroup

Precise regularity estimates on diffusion semigroups are more than a mere theoretical curiosity. They play a fundamental role in deducing sharp error bounds for higher-order particle methods. In this

Hypoelliptic non-homogeneous diffusions

Abstract. Let be a time dependent second order operator, written in usual or Hörmander form. We study the regularity of the law of the associated non-homogeneous (time dependent) diffusion process,

NONAUTONOMOUS KOLMOGOROV PARABOLIC EQUATIONS WITH UNBOUNDED COEFFICIENTS

We study a class of elliptic operators A with unbounded coeffi- cients defined in I × R d for some unbounded interval IR. We prove that, for any s 2 I, the Cauchy problem u(s,·) = f 2 Cb(R d ) for

Irreversible Langevin samplers and variance reduction: a large deviations approach

In order to sample from a given target distribution (often of Gibbs type), the Monte Carlo Markov chain method consists of constructing an ergodic Markov process whose invariant measure is the target

Ergodic properties of Markov processes

In these notes we discuss Markov processes, in particular stochastic differential equations (SDE) and develop some tools to analyze their long-time behavior. There are several ways to analyze such

Variance reduction for irreversible Langevin samplers and diffusion on graphs

In recent papers it has been demonstrated that sampling a Gibbs distribution from an appropriate time-irreversible Langevin process is, from several points of view, advantageous when compared to

On the convergence rates of a general class of weak approximations of SDEs

Abstract In this paper, the convergence analysis of a class of weak approximations of solutions of stochastic differential equations is presented. This class includes recent approximations such as

Orbits of families of vector fields and integrability of distributions

Let D be an arbitrary set of Cc vector fields on the Cc manifold M. It is shown that the orbits of D are C' submanifolds of M, and that, moreover, they are the maximal integral submanifolds of a
...