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Long-time $L^p$ Wasserstein contraction for diffusion processes without global dissipativity

Pierre Monmarché

Abstract

The fact that a Markov diffusion semi-group on $\mathbb R^d$ contracts the $L^p$ Wasserstein distance, which has been extensively used to establish uniform-in-time stability estimates (e.g. with respect to numerical discretization errors), is a well-studied question in the case where the distances are in fact deterministically contracted by the drift (global dissipativity condition) or in the case $p=1$ (with reflection couplings). This work focuses on the non-globally dissipative case with $p>1$. This situation was previously considered in \cite{MonmarcheBruit}, but only for elliptic processes, and with a restriction on the diffusivity coefficient (which had to be large enough). Here, we extend this analysis to non-elliptic processes and provide sharper conditions to get contractions along synchronous coupling, including negative results, lower bounds and a characterization (at least in dimension 1) in terms of the maximal eigenvalue of a Feynman-Kac operator.

Long-time $L^p$ Wasserstein contraction for diffusion processes without global dissipativity

Abstract

The fact that a Markov diffusion semi-group on contracts the Wasserstein distance, which has been extensively used to establish uniform-in-time stability estimates (e.g. with respect to numerical discretization errors), is a well-studied question in the case where the distances are in fact deterministically contracted by the drift (global dissipativity condition) or in the case (with reflection couplings). This work focuses on the non-globally dissipative case with . This situation was previously considered in \cite{MonmarcheBruit}, but only for elliptic processes, and with a restriction on the diffusivity coefficient (which had to be large enough). Here, we extend this analysis to non-elliptic processes and provide sharper conditions to get contractions along synchronous coupling, including negative results, lower bounds and a characterization (at least in dimension 1) in terms of the maximal eigenvalue of a Feynman-Kac operator.
Paper Structure (23 sections, 15 theorems, 168 equations, 1 figure)

This paper contains 23 sections, 15 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Motivation and difficulties
  3. Settings and a general result
  4. Two negative results
  5. Criteria for long-time contraction
  6. An asymptotic criterion for contraction
  7. Numerical experiments
  8. A constructive non-asymptotic criterion
  9. Examples
  10. The elliptic case
  11. Kinetic Langevin process
  12. Colored noise diffusion
  13. Morris-Lecar neuron model
  14. On the general results
  15. Sharper analysis of the Lyapunov exponent
  16. ...and 8 more sections

Key Result

Proposition 1

Under Assumption Assum:basic, for all $p\geqslant 1$ and $t\geqslant 0$, where, in the right hand side, $(X_t,Y_t)_{t\geqslant 0}$ are two solutions of eq:EDS_X (driven by the same Brownian motion $B$) with $(X_0,Y_0)=(x,y)$. In particular, for any $t\geqslant 0$, given two solutions of eq:EDS_X (with the same Brownian motion), As a consequence, for all $p\geqslant 1$ and $t\geqslant 0$, $\beta_

Figures (1)

  • Figure 1: Top: potentials $U_1$ (left) and $U_2$ (right). Bottom: estimation of $\mathcal{J}(p\eta)/p$ for \ref{['locXU']} (left with $U_1$, right with $U_2$), as a function of $\theta^2\in[0.1,2]$ and $p\in[1,3]$ (on the right graph, the color bar is capped in $[-4,4]$).

Theorems & Definitions (39)

  • Proposition 1
  • Proposition 2
  • Remark 1
  • Remark 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Remark 3
  • Remark 4
  • ...and 29 more