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Limiting Behavior of Randomly Perturbed Diffusions with Invariant Repelling Surfaces

Leonid Koralov, Chenglin Liu

TL;DR

The paper analyzes small non-degenerate perturbations of degenerate diffusions in $\mathbb{R}^d$ that preserve repelling invariant surfaces, deriving precise near-boundary invariant-density asymptotics and a multiscale metastable framework. It introduces a boundary-layer expansion of the generator, identifies the critical exponents $\gamma_k$ from a surface-local spectral problem, and shows that on time scales $\varepsilon^{\gamma_{k-1}}\ll t(\varepsilon)\ll\varepsilon^{\gamma_k}$ the diffusion's law converges to a linear combination of the unperturbed invariant measures $\mu_j$, with coefficients $c_j(i,k)$ depending only on the generator $L$. The results extend to a tree-structured domain graph, where the metastable distributions are determined by a semi-Markov reduction on clusters, and the largest time-scale limit is a fixed convex combination $\sum_j c_j\mu_j$ independent of the starting domain. The work provides explicit asymptotics for invariant densities, transition probabilities, and exit times, yielding a concrete mechanism for metastability in degenerate diffusions with invariant surfaces and a procedure to compute the coefficients from $L$ alone.

Abstract

We study small perturbations of diffusion processes in $\mathbb{R}^d$ that leave invariant a finite collection of hypersurfaces. Each surface is assumed to be repelling for the unperturbed process, and the unperturbed motion on each of the surfaces is assumed to be ergodic. These surfaces separate the space into a finite number of domains, each of which carries an invariant measure of the unperturbed process. We describe the asymptotics of the densities of the invariant measures near the invariant surfaces. We then describe the asymptotic behavior of the perturbed process: at different time scales (depending on the size of the perturbation), metastable distributions are described in terms of linear combinations of the ergodic invariant measures of the unperturbed system. The coefficients in the linear combination depend on the time scale but are shown not to depend on the perturbation.

Limiting Behavior of Randomly Perturbed Diffusions with Invariant Repelling Surfaces

TL;DR

The paper analyzes small non-degenerate perturbations of degenerate diffusions in that preserve repelling invariant surfaces, deriving precise near-boundary invariant-density asymptotics and a multiscale metastable framework. It introduces a boundary-layer expansion of the generator, identifies the critical exponents from a surface-local spectral problem, and shows that on time scales the diffusion's law converges to a linear combination of the unperturbed invariant measures , with coefficients depending only on the generator . The results extend to a tree-structured domain graph, where the metastable distributions are determined by a semi-Markov reduction on clusters, and the largest time-scale limit is a fixed convex combination independent of the starting domain. The work provides explicit asymptotics for invariant densities, transition probabilities, and exit times, yielding a concrete mechanism for metastability in degenerate diffusions with invariant surfaces and a procedure to compute the coefficients from alone.

Abstract

We study small perturbations of diffusion processes in that leave invariant a finite collection of hypersurfaces. Each surface is assumed to be repelling for the unperturbed process, and the unperturbed motion on each of the surfaces is assumed to be ergodic. These surfaces separate the space into a finite number of domains, each of which carries an invariant measure of the unperturbed process. We describe the asymptotics of the densities of the invariant measures near the invariant surfaces. We then describe the asymptotic behavior of the perturbed process: at different time scales (depending on the size of the perturbation), metastable distributions are described in terms of linear combinations of the ergodic invariant measures of the unperturbed system. The coefficients in the linear combination depend on the time scale but are shown not to depend on the perturbation.
Paper Structure (9 sections, 27 theorems, 196 equations)

This paper contains 9 sections, 27 theorems, 196 equations.

Table of Contents

  1. Introduction and main results
  2. Behavior of the processes TEXT and TEXT and of the corresponding operators near TEXT
  3. The structure of the invariant measure for TEXT in a single domain
  4. Transition probabilities and exit times when exiting one domain
  5. Transition probabilities for TEXT
  6. Exit time of TEXT to surfaces
  7. Transition probabilities and exit times starting from the surfaces
  8. Metastable behavior of the diffusion process
  9. Proof of Theorem \ref{['main theorem']}

Key Result

Theorem 1

Suppose that Assumptions (a)-(e) hold and all surfaces $S_k$, $1 \leq k \leq m$, are repelling (with $\gamma_k < 0$). There exist non-negative constants $c_j(i,k)$ that are determined by the coefficients of the operator $L$ (i.e., by the vector fields $v_0, v_1,...,v_d$) but do not depend on the per uniformly for all $x$ in a compact set in $D_i$, for all $\varepsilon^{\gamma_{k-1}}\ll t(\varepsil

Theorems & Definitions (34)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 24 more