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Large deviation principle for the stationary solutions of stochastic functional differential equations with infinite delay

Yong Liu, Bin Tang

TL;DR

This work establishes a uniform Large Deviation Principle for the solution maps of stochastic functional differential equations with infinite delay under small noise, and derives an LDP for the corresponding stationary solutions and their invariant measures. The authors develop a tailored weak convergence approach, including a modified Boué-Dupuis variational representation and controlled SFDEs, to prove uniform Laplace principles on the path space $\mathscr{C}_r$. Central contributions include the well-posedness and compactness of skeleton equations, the uniform LDP for solution maps via the rate functions $\mathcal{I}_{t_0,\xi}$ and $\mathcal{I}_{-\infty}$, and the LDP for stationary solutions with rate function $\mathcal{I}_{-\infty}$, plus contraction to obtain the LDP for invariant measures with rate function $\mathcal{I}'$. The results illuminate the long-time behavior and rare-event probabilities for SFDEs with infinite delay, extending existing LDP frameworks from finite-delay settings to this broader, infinite-dimensional context.

Abstract

We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness of the corresponding stationary solutions. Second, by the weak convergence approach, we show the uniform large deviation principle for the solution maps, and then prove the LDP for stationary solutions. Furthermore, we obtain the LDP for invariant measures of SFDEs through the LDP for stationary solutions and the contraction principle.

Large deviation principle for the stationary solutions of stochastic functional differential equations with infinite delay

TL;DR

This work establishes a uniform Large Deviation Principle for the solution maps of stochastic functional differential equations with infinite delay under small noise, and derives an LDP for the corresponding stationary solutions and their invariant measures. The authors develop a tailored weak convergence approach, including a modified Boué-Dupuis variational representation and controlled SFDEs, to prove uniform Laplace principles on the path space . Central contributions include the well-posedness and compactness of skeleton equations, the uniform LDP for solution maps via the rate functions and , and the LDP for stationary solutions with rate function , plus contraction to obtain the LDP for invariant measures with rate function . The results illuminate the long-time behavior and rare-event probabilities for SFDEs with infinite delay, extending existing LDP frameworks from finite-delay settings to this broader, infinite-dimensional context.

Abstract

We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness of the corresponding stationary solutions. Second, by the weak convergence approach, we show the uniform large deviation principle for the solution maps, and then prove the LDP for stationary solutions. Furthermore, we obtain the LDP for invariant measures of SFDEs through the LDP for stationary solutions and the contraction principle.
Paper Structure (10 sections, 17 theorems, 98 equations)

This paper contains 10 sections, 17 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminary and main results
  3. SFDE and stationary solution
  4. SFDE
  5. Stationary solution
  6. Large Deviations
  7. Skeleton equations
  8. Uniform Large Deviation for solution map
  9. Large Deviations for stationary solution
  10. Analysis tool

Key Result

Theorem 2.2

For any $\epsilon>0$, there is a pathwise unique stationary solution $\mathcal{Y}^{\epsilon} \in L^2(\Omega; \mathscr{C}_r)$ for crude cocycle $(U^{\epsilon},\theta)$. For any $t \in \mathbb{R}$ and $\tau \in (-\infty,0]$, $\mathcal{Y}^{\epsilon}(t)$$\mathbb{P}$-a.s. satisfies and has a uniform bound specifically $\sup_{t \in \mathbb{R}} \mathbb{E} \| \mathcal{Y}^{\epsilon}(t) \|_{r}^2 < +\infty$

Theorems & Definitions (34)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: Rate Function
  • Definition 2.4: Large Deviation Principle
  • Definition 2.5: Freidlin-Wentzell uniform LDP
  • Definition 2.6: Laplace Principle
  • Definition 2.7: Uniform Laplace Principle
  • Theorem 2.8
  • Theorem 2.9
  • Corollary 2.10
  • ...and 24 more