[Paper Review] Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence

Tony Lelièvre, Mouad Ramil+1 more|arXiv (Cornell University)|Jan 28, 2021

Diffusion and Search Dynamics44 references20 citations

TL;DR

This paper establishes the existence, uniqueness, and exponential convergence to a quasi-stationary distribution (QSD) for the Langevin process in cylindrical domains $D = \mathcal{O} \times \mathbb{R}^d$, where $\mathcal{O}$ is a bounded $\mathcal{C}^2$ domain. By proving compactness of the absorbed semigroup and applying the Krein-Rutman theorem, it shows the QSD arises as the principal eigenfunction of the infinitesimal generator with Dirichlet boundary conditions, ensuring long-time convergence of conditioned processes at an exponential rate.

ABSTRACT

Consider the Langevin process, described by a vector (position,momentum) in $\\mathbb{R}^{d}\ imes\\mathbb{R}^d$. Let $\\mathcal O$ be a $\\mathcal{C}^2$ open bounded and connected set of $\\mathbb{R}^d$. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain $D:=\\mathcal{O}\ imes\\mathbb{R}^d$. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on $D$. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD.

Motivation & Objective

• To establish the existence and uniqueness of a quasi-stationary distribution (QSD) for the Langevin process in bounded cylindrical domains $D = \mathcal{O} \times \mathbb{R}^d$.
• To address the lack of results for the hypoelliptic Langevin dynamics, which is not elliptic and has unbounded momentum space.
• To prove exponential convergence of the process conditioned on non-absorption to the QSD, ensuring long-time statistical stability within metastable states.
• To provide a spectral characterization of the QSD via the principal eigenvalue and eigenfunction of the infinitesimal generator with absorbing boundary conditions.

Proposed method

• Prove compactness of the semigroup associated with the absorbed Langevin process on $D$ using transition density bounds from prior work.
• Apply the Krein-Rutman theorem to the infinitesimal generator of the absorbed process to establish a principal eigenvalue and positive eigenfunction.
• Characterize the QSD as the normalized eigenfunction corresponding to the principal eigenvalue, ensuring uniqueness and spectral gap.
• Use spectral decomposition to derive exponential convergence rates of the conditioned process toward the QSD.
• Leverage probabilistic coupling and semigroup properties to bound the total variation distance between the conditioned law and the QSD.
• Validate the convergence rate via estimates involving the spectral gap $\lambda_0$ and a parameter $\alpha \in [0, \alpha^*)$.

Experimental results

Research questions

• RQ1 Does a unique quasi-stationary distribution exist for the Langevin process in cylindrical domains $D = \mathcal{O} \times \mathbb{R}^d$ where $\mathcal{O}$ is a bounded $\mathcal{C}^2$ domain?
• RQ2 Can the long-time behavior of the Langevin process, conditioned on non-absorption, be characterized by a unique limiting distribution?
• RQ3 What is the spectral structure of the infinitesimal generator of the absorbed Langevin process, and how does it relate to the QSD?
• RQ4 Is the convergence of the conditioned process to the QSD exponentially fast, and what is the rate in terms of spectral data?
• RQ5 How does the hypoelliptic nature of the Langevin dynamics affect the existence and properties of the QSD compared to elliptic diffusions?

Key findings

• The semigroup of the absorbed Langevin process on $D$ is compact, a key step toward spectral analysis.
• A unique quasi-stationary distribution $\mu$ exists for the Langevin process on $D$, characterized as the normalized eigenfunction of the infinitesimal generator associated with the principal eigenvalue $\lambda_0$.
• The QSD attracts all initial distributions on $D$ at an exponential rate, with the convergence rate controlled by $\lambda_0 + \alpha$ for any $\alpha \in [0, \alpha^*)$.
• The total variation distance between the conditioned law at time $t$ and the QSD is bounded by $C_{\alpha} \|f\|_{\infty} \mathrm{e}^{-\alpha t}$, with $C_{\alpha}$ depending on the initial measure and the spectral gap.
• The QSD is the unique solution to an eigenvalue problem involving the infinitesimal generator with Dirichlet boundary conditions on $\partial D$
• The convergence result holds uniformly over initial probability measures, with explicit dependence on the initial mass of the Perron-Frobenius eigenfunction $\phi$.