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Metastability in Parabolic Equations and Diffusion Processes with a Small Parameter

Mark Freidlin, Leonid Koralov

Abstract

We study diffusion processes in $\mathbb{R}^d$ that leave invariant a finite collection of manifolds (surfaces or points) in $\mathbb{R}^d$ and small perturbations of such processes. Assuming certain ergodic properties at and near the invariant surfaces, we describe the rate at which the process gets attracted to or repelled from the surface, based on the local behavior of the coefficients. For processes that include, additionally, a small non-degenerate perturbation, we describe the metastable behavior. Namely, by allowing the time scale to depend on the size of the perturbation, we observe different asymptotic distributions of the process at different time scales. Stated in PDE terms, the results provide the asymptotics, at different time scales, for the solution of the parabolic Cauchy problem when the operator that degenerates on a collection of surfaces is perturbed by a small non-degenerate term. This asymptotic behavior switches at a finite number of time scales that are calculated and does not depend on the perturbation.

Metastability in Parabolic Equations and Diffusion Processes with a Small Parameter

Abstract

We study diffusion processes in that leave invariant a finite collection of manifolds (surfaces or points) in and small perturbations of such processes. Assuming certain ergodic properties at and near the invariant surfaces, we describe the rate at which the process gets attracted to or repelled from the surface, based on the local behavior of the coefficients. For processes that include, additionally, a small non-degenerate perturbation, we describe the metastable behavior. Namely, by allowing the time scale to depend on the size of the perturbation, we observe different asymptotic distributions of the process at different time scales. Stated in PDE terms, the results provide the asymptotics, at different time scales, for the solution of the parabolic Cauchy problem when the operator that degenerates on a collection of surfaces is perturbed by a small non-degenerate term. This asymptotic behavior switches at a finite number of time scales that are calculated and does not depend on the perturbation.
Paper Structure (13 sections, 28 theorems, 178 equations)

This paper contains 13 sections, 28 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Asymptotic behavior of the unperturbed processes
  3. Structure of the generator near an invariant surface
  4. Transition probabilities near an invariant surface
  5. Limiting distribution of the unperturbed process
  6. Asymptotic behavior of the perturbed process
  7. Transition probabilities near an invariant surface
  8. Transition times near an invariant surface
  9. Distribution of the process as it exits a neighborhood of an invariant surface
  10. Metastable distributions for the perturbed process
  11. Asymptotics of transition probabilities between different attracting surfaces
  12. The case when at least one of the invariant surfaces is attracting
  13. The case when all the components of the boundary are repelling

Key Result

Theorem 1.1

Let all invariant manifolds have dimensions $d_k < d-1$. (See the discussion below for the case $d_k = d-1$.) Under assumptions (a)-(e) on the coefficients of the operator $L^\varepsilon$, the asymptotics of solutions to the initial-boundary value problem (direq2) is as follows. (a) If at least one (b) If all the components of the boundary are repelling (i.e., $\gamma_1 < 0$), then, for $x \in D$

Theorems & Definitions (28)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Lemma 3.1
  • ...and 18 more