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Exit-problem for a class of non-Markov processes with path dependency

Ashot Aleksian, Aline Kurtzmann, Julian Tugaut

TL;DR

This work analyzes the exit-time of a self-interacting diffusion (SID) with path-dependent drift from a bounded domain $G$ in the small-noise regime, without requiring convexity of the confinement or interaction potentials. The authors establish a Large Deviation Principle (LDP) for SID under weak regularity, then adapt Freidlin–Wentzell techniques to prove a Kramers-type law for the exit-time, characterized by the effective-potential height $H= obreak igl( obracket U_a(z)-U_a(a)igr)$ minimized over the boundary $ abla G$, where $U_a=V+F(ullet-a)$. A key contribution is handling generalized initial conditions via a contraction argument and proving exit-location results: exits concentrate on boundary points attaining the minimum of $U_a$. Overall, the results extend metastability analyses of self-interacting diffusions to nonconvex landscapes and path-dependent drifts, offering a rigorous framework for small-noise exit behavior and exit locations in SID systems.

Abstract

We study the exit-time of a self-interacting diffusion from an open domain $G \subset \mathbb{R}^d$. In particular, we consider the equation $d{X_t} = - \left( \nabla V(X_t) + \frac{1}{t}\int_0^t\nabla F (X_t - X_s)d{s} \right) d{t} + σd{W_t}.$ We are interested in the small-noise ($σ\to 0$) behaviour of the exit-time from the potentials' domain of attraction. In this work rather weak assumptions on the potentials $V$ and $F$, and on the domain $G$ are considered. In particular, we do not assume $V$ nor $F$ to be either convex or concave, which covers a wide range of self-attracting and self-repelling stochastic processes possibly moving in a complex multi-well landscape. The Large Deviation Principle for the Self-interacting diffusion with generalized initial conditions is established. The main result of the paper states that, under some assumptions on the potentials $V$ and $F$, and on the domain $G$, the Kramers' type law for the exit-time holds. Finally, we provide a result concerning the exit-location of the diffusion.

Exit-problem for a class of non-Markov processes with path dependency

TL;DR

This work analyzes the exit-time of a self-interacting diffusion (SID) with path-dependent drift from a bounded domain in the small-noise regime, without requiring convexity of the confinement or interaction potentials. The authors establish a Large Deviation Principle (LDP) for SID under weak regularity, then adapt Freidlin–Wentzell techniques to prove a Kramers-type law for the exit-time, characterized by the effective-potential height minimized over the boundary , where . A key contribution is handling generalized initial conditions via a contraction argument and proving exit-location results: exits concentrate on boundary points attaining the minimum of . Overall, the results extend metastability analyses of self-interacting diffusions to nonconvex landscapes and path-dependent drifts, offering a rigorous framework for small-noise exit behavior and exit locations in SID systems.

Abstract

We study the exit-time of a self-interacting diffusion from an open domain . In particular, we consider the equation We are interested in the small-noise () behaviour of the exit-time from the potentials' domain of attraction. In this work rather weak assumptions on the potentials and , and on the domain are considered. In particular, we do not assume nor to be either convex or concave, which covers a wide range of self-attracting and self-repelling stochastic processes possibly moving in a complex multi-well landscape. The Large Deviation Principle for the Self-interacting diffusion with generalized initial conditions is established. The main result of the paper states that, under some assumptions on the potentials and , and on the domain , the Kramers' type law for the exit-time holds. Finally, we provide a result concerning the exit-location of the diffusion.
Paper Structure (32 sections, 18 theorems, 186 equations, 6 figures)

This paper contains 32 sections, 18 theorems, 186 equations, 6 figures.

Table of Contents

  1. Introduction
  2. State of the art
  3. Some remarks on dynamics and initial condition
  4. Outline of the paper
  5. List of notations
  6. Large Deviation Principle
  7. Establishing the LDP for the SID
  8. Results related to the LDP
  9. Compactness results
  10. Proof of the main theorem
  11. Auxiliary results
  12. Initial descent to the point of attraction a
  13. Descent of the deterministic process towards a
  14. Attraction of the stochastic process towards a
  15. Behaviour in the annulus between Brho(a) and boundary of G
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let Assumptions A-1 and A-2 be fulfilled. Let the process $X^\sigma$ be the unique strong solution of the system eq:SID_main_sys. Let $\tau_G^\sigma := \inf \{t: X_t^\sigma \notin G\}$ denote the first time when $X^\sigma$ exits the domain $G$. Let $H:= \inf_{x \in \partial G} \{U_a(x) - U_a(a)\}$ b

Figures (6)

  • Figure 1: Examples of possible $V$, $F$, and $G$ in dimension $d = 1$, with $U_a$ being the effective potential.
  • Figure 2: Illustration of the relationship between some objects used in the proof, namely $\mathbf{x}$, $\Phi^\mathbf{x}$, $\mathbb{B}_{\delta_\mathbf{x}}(\mathbf{x})$, and $\bigcup_{\mathbf{y} \in \mathbb{B}_{\delta_\mathbf{x}}(\mathbf{x})} \Phi^{\mathbf{y}}$.
  • Figure 3: Illustration of the definitions of $\tau_k$ and $\theta_k$ for $d = 2$.
  • Figure 4: Domain $G$ with level sets of $U_a = V + F(\cdot - a) - V(a)$. $L_{H} = \{x \in \mathbb{R}^d: U_a(x) = H\}$ is the smallest level set that touches the boundary $\partial G$
  • Figure 5: Dynamics of $\boldsymbol{X}^{0}$
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 1.1
  • Definition 1
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 18 more