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Bounding Escape Rates and Approximating Quasi-Stationary Distributions of Brownian Dynamics

Jason J. Bramburger

TL;DR

The paper develops a computational framework to bound the principal eigenvalue $\lambda_0$ of the Brownian-generator for dynamics in a bounded domain, linking $\lambda_0$ to the exponential escape rate and the quasi-stationary distribution $u_0$. By transforming to the Schrödinger-type operator $\mathcal{H}=\sigma\Delta - U(x)$ and employing a pointwise dual relaxation with null Lagrangians, it yields convergent analytical and SOS-based lower bounds on $\lambda_0$, with upper bounds obtained from Rayleigh quotients using near-optimal eigenfunction information. For polynomial $U$ and semialgebraic domains, the SOS program provides an increasing sequence of bounds $\lambda_\nu^{\mathrm{SOS}}$ that converge to $\lambda_0$ (under Dirichlet boundary conditions), and numerical demonstrations on a double-well potential and a unit ball show tight bounds and accurate approximations to the leading eigenfunction $u_0$ and the QSD. The method, implemented via YALMIP/MOSEK, is particularly effective in low dimensions and offers a pathway to rigorous numerics and applications to large-deviation analyses, with potential as a preconditioner for computer-assisted proofs.

Abstract

Throughout physics Brownian dynamics are used to describe the behaviour of molecular systems. When the Brownian particle is confined to a bounded domain, a particularly important question arises around determining how long it takes the particle to encounter certain regions of the boundary from which it can escape. Termed the first passage time, it sets the natural timescale of the chemical, biological, and physical processes that are described by the stochastic differential equation. Probabilistic information about the first passage time can be studied using spectral properties of the deterministic generator of the stochastic process. In this work we introduce a framework for bounding the leading eigenvalue of the generator which determines the exponential rate of escape of the particle from the domain. The method employs sum-of-squares programming to produce nearly sharp numerical upper and lower bounds on the leading eigenvalue, while also giving good numerical approximations of the associated leading eigenfunction, the quasi-stationary distribution of the process. To demonstrate utility, the method is applied to prototypical low-dimensional problems from the literature.

Bounding Escape Rates and Approximating Quasi-Stationary Distributions of Brownian Dynamics

TL;DR

The paper develops a computational framework to bound the principal eigenvalue of the Brownian-generator for dynamics in a bounded domain, linking to the exponential escape rate and the quasi-stationary distribution . By transforming to the Schrödinger-type operator and employing a pointwise dual relaxation with null Lagrangians, it yields convergent analytical and SOS-based lower bounds on , with upper bounds obtained from Rayleigh quotients using near-optimal eigenfunction information. For polynomial and semialgebraic domains, the SOS program provides an increasing sequence of bounds that converge to (under Dirichlet boundary conditions), and numerical demonstrations on a double-well potential and a unit ball show tight bounds and accurate approximations to the leading eigenfunction and the QSD. The method, implemented via YALMIP/MOSEK, is particularly effective in low dimensions and offers a pathway to rigorous numerics and applications to large-deviation analyses, with potential as a preconditioner for computer-assisted proofs.

Abstract

Throughout physics Brownian dynamics are used to describe the behaviour of molecular systems. When the Brownian particle is confined to a bounded domain, a particularly important question arises around determining how long it takes the particle to encounter certain regions of the boundary from which it can escape. Termed the first passage time, it sets the natural timescale of the chemical, biological, and physical processes that are described by the stochastic differential equation. Probabilistic information about the first passage time can be studied using spectral properties of the deterministic generator of the stochastic process. In this work we introduce a framework for bounding the leading eigenvalue of the generator which determines the exponential rate of escape of the particle from the domain. The method employs sum-of-squares programming to produce nearly sharp numerical upper and lower bounds on the leading eigenvalue, while also giving good numerical approximations of the associated leading eigenfunction, the quasi-stationary distribution of the process. To demonstrate utility, the method is applied to prototypical low-dimensional problems from the literature.

Paper Structure

This paper contains 15 sections, 2 theorems, 61 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. First Passage Times
  3. Bounds on the Principal Eigenvalue
  4. Lower Bounds on the Principal Eigenvalue
  5. Lower Bounds on the Principal Eigenvalue
  6. Relaxation to a Sum-of-Squares Program
  7. Upper Bounds on the Principal Eigenvalue
  8. Numerical Demonstrations
  9. Escape From a Double Well Potential
  10. Brownian Motion in a Ball
  11. Fully Absorbing Boundary
  12. Partially Absorbing Boundary
  13. Two Exit Regions.
  14. One Exit Region.
  15. Discussion

Key Result

Theorem 3.1

chernyavsky2023convex Let $\Omega$ be an open, bounded, and Lipschitz domain, $U:\mathbb{R}^d \to \mathbb{R}$ a smooth function, $\sigma > 0$, and $\Gamma_A = \partial \Omega$. Then, $\lambda_0 = \lambda_\mathrm{pdr}$. In particular, there exists $\lambda^k \in \mathbb{R}$ and $f^k\in C(\overline\Om for all $k \geq 1$ and $\lambda^k \nearrow \lambda_0$ as $k \to \infty$.

Figures (3)

  • Figure 1: Left: Numerical upper and lower bounds on the principal eigenvalue of the operator \ref{['DoubleWellOperator']} with $L = 3$ using degree $\nu = 6,8,10$ polynomials. Right: Approximations of the principal eigenfunction of the operator \ref{['DoubleWellOperator']} for $\alpha = 0,1$ using the degree $10$ polynomials that provide the upper bounds at these parameter values.
  • Figure 2: Left: The unit ball in $\mathbb{R}^2$ with exits as in \ref{['EscapeLocations']} symmetrically centered at $(\pm 1,0)$ on the boundary. Right: A single exit centered at $(1,0)$ on the boundary.
  • Figure 3: Surface plots of the approximations of the principal eigenfunctions for the Laplacian with partially absorbing boundaries given by \ref{['EscapeLocations']} with $(k,r) = (2,0.5)$ (left) and $(k,r) = (1,0.5)$ (right).

Theorems & Definitions (4)

  • Theorem 3.1
  • proof : Sketch of Proof of Theorem \ref{['thm:PDRConv']}
  • Remark 1
  • Theorem 3.2