[Paper Review] Almost sure contraction for diffusions on $\\mathbb R^d$. Application to generalised Langevin diffusions

Pierre Monmarché|arXiv (Cornell University)|Sep 22, 2020

Markov Chains and Monte Carlo Methods34 references4 citations

TL;DR

This paper establishes almost sure contraction for diffusions on $\mathbb{R}^d$ with constant diffusion matrix, proving that deterministic drift contractivity is equivalent to exponential contraction of $\mathcal{W}_p$ Wasserstein distances for all $p \in [1, \infty]$. The result is applied to generalized Langevin diffusions, showing contraction under convex potential and high friction, extending prior results beyond reversible or hypoelliptic settings.

ABSTRACT

In the case of diffusions on $\\mathbb R^d$ with constant diffusion matrix, without assuming reversibility nor hypoellipticity, we prove that the contractivity of the deterministic drift is equivalent to the constant rate contraction of Wasserstein distances $\\mathcal W_p$, $p\\in[1,\\infty]$. It also implies concentration inequalities for ergodic means of the process. Such a contractivity property is then established for some non-equilibrium chains of anharmonic oscillators and for some generalised Langevin diffusions when the potential is convex with bounded Hessian and the friction is sufficiently high. This extends previous known results for the usual (kinetic) Langevin diffusion.

Motivation & Objective

• To establish a general equivalence between deterministic drift contractivity and exponential contraction of $\mathcal{W}_p$ Wasserstein distances for diffusions on $\mathbb{R}^d$ with constant diffusion matrix.
• To extend known contraction results beyond reversible or hypoelliptic diffusions, particularly to non-equilibrium generalized Langevin processes.
• To provide conditions under which generalized Langevin diffusions exhibit almost sure contraction, even without reversibility or hypoellipticity.
• To demonstrate that convexity of the potential and sufficiently high friction are sufficient for contraction, generalizing previous results on overdamped and kinetic Langevin diffusions.

Proposed method

• Uses coupling arguments and semigroup interpolation techniques to link gradient contraction to Wasserstein distance contraction.
• Applies the Bakry-Émery curvature framework in a modified form, incorporating a vertical carré du champ $\Gamma^Z$ to handle degenerate diffusions.
• Employs a modified generator $\mathcal{L}^*$ and duality arguments to analyze the adjoint semigroup, enabling contraction estimates.
• Introduces a norm $\|\cdot\|_K$ associated with a positive definite matrix $K$ to define a Riemannian structure compatible with the dynamics.
• Derives exponential decay estimates for $\mathcal{W}_{K,2}$ distances via spectral bounds on the adjoint semigroup $R_t$, using matrix inequalities and operator norms.
• Applies results from hypocoercivity and duality to show $\|P_t f\|_{L^2(\mu_\infty)} \leq \sqrt{\eta_t} \|f\|_{L^2(\mu_\infty)}$ with $\eta_t = 3|N^{-1}||N|e^{-2\rho t}$, implying $L^2$ decay.

Experimental results

Research questions

• RQ1 Is the contractivity of the deterministic drift equivalent to exponential contraction of $\mathcal{W}_p$ distances for general diffusions on $\mathbb{R}^d$ without assuming reversibility or hypoellipticity?
• RQ2 Can generalized Langevin diffusions with convex potentials and high friction exhibit almost sure contraction, even when the process is non-reversible?
• RQ3 What conditions on the potential $U$ and friction parameter $\gamma$ ensure exponential contraction of Wasserstein distances in non-equilibrium diffusions?
• RQ4 How does the absence of Bakry-Émery curvature ($\rho = -\infty$) affect the contraction properties of kinetic diffusions?
• RQ5 Can the contraction of $\mathcal{W}_p$ distances be established via duality and modified generators when standard curvature conditions fail?

Key findings

• The contractivity of the deterministic drift is equivalent to exponential contraction of $\mathcal{W}_p$ distances for all $p \in [1, \infty]$, under constant diffusion matrix and without requiring reversibility or hypoellipticity.
• For generalized Langevin diffusions, contraction holds when the potential $U$ is convex with bounded Hessian and the friction $\gamma$ is sufficiently high.
• The contraction rate $\rho > 0$ is explicitly bounded in terms of the Hessian bounds of $U$, the friction $\gamma$, and the matrix $N$ related to the dynamics.
• The paper establishes concentration inequalities for ergodic means of the process under the same conditions, linking pathwise contraction to statistical concentration.
• The contraction result is robust under duality: the adjoint semigroup $R_t$ satisfies $\|P_t f\|_{L^2(\mu_\infty)} \leq \sqrt{\eta_t} \|f\|_{L^2(\mu_\infty)}$ with $\eta_t = 3|N^{-1}||N|e^{-2\rho t}$, implying $L^2$ decay.
• The method extends to non-reversible processes by introducing a vertical carré du champ $\Gamma^Z$, enabling contraction estimates even when standard Bakry-Émery curvature is $-\infty$.