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Quasi-ergodic theorems for Feynman-Kac semigroups and large deviation for additive functionals

Daehong Kim, Takara Tagawa, Aurélien Velleret

TL;DR

The paper analyzes the long-time behavior of additive functionals for symmetric Markov processes under non-local Feynman--Kac bias and extinction. By leveraging ground-state transforms and Fukushima ergodic theory, it derives quasi-ergodic limits for both continuous and jump components, expressed through the ground state $\phi_0$ and the jump measure $\mathcal{J}_{\phi_0}$, and proves a conditional weak law of large numbers. A large deviation principle for the mean of the additive functionals is established via a spectrally-informed rate function derived from a holomorphic family $\lambda_0(\theta)$ and its derivative, connecting spectral data to deviations of $A_t^{V,F}$. The results apply under explicit Kato-class and intrinsic ultracontractivity assumptions, with concrete examples including symmetric Lévy processes and absorbing domains, highlighting the framework's reach for non-local Feynman--Kac penalizations in quasi-stationary settings.

Abstract

We study the long-time behavior of an additive functional that takes into account the jumps of a symmetric Markov process. This process is assumed to be observed through a biased observation scheme that includes the survival to events of extinction and the Feynman-Kac weight by another similar additive functional. Under conditioning for the convergence to a quasi-stationary distribution and for two-sided estimates of the Feynmac-Kac semigroup to be obtained, we shall discuss general assumptions on the symmetric Markov process. For the law of additive functionals, we will prove a quasi-ergodic theorem, namely a conditional version of the ergodic theorem and a conditional functional weak law of large numbers. As an application, we also establish a large deviation principle for the mean ratio of additive functionals.

Quasi-ergodic theorems for Feynman-Kac semigroups and large deviation for additive functionals

TL;DR

The paper analyzes the long-time behavior of additive functionals for symmetric Markov processes under non-local Feynman--Kac bias and extinction. By leveraging ground-state transforms and Fukushima ergodic theory, it derives quasi-ergodic limits for both continuous and jump components, expressed through the ground state and the jump measure , and proves a conditional weak law of large numbers. A large deviation principle for the mean of the additive functionals is established via a spectrally-informed rate function derived from a holomorphic family and its derivative, connecting spectral data to deviations of . The results apply under explicit Kato-class and intrinsic ultracontractivity assumptions, with concrete examples including symmetric Lévy processes and absorbing domains, highlighting the framework's reach for non-local Feynman--Kac penalizations in quasi-stationary settings.

Abstract

We study the long-time behavior of an additive functional that takes into account the jumps of a symmetric Markov process. This process is assumed to be observed through a biased observation scheme that includes the survival to events of extinction and the Feynman-Kac weight by another similar additive functional. Under conditioning for the convergence to a quasi-stationary distribution and for two-sided estimates of the Feynmac-Kac semigroup to be obtained, we shall discuss general assumptions on the symmetric Markov process. For the law of additive functionals, we will prove a quasi-ergodic theorem, namely a conditional version of the ergodic theorem and a conditional functional weak law of large numbers. As an application, we also establish a large deviation principle for the mean ratio of additive functionals.
Paper Structure (10 sections, 14 theorems, 144 equations)

This paper contains 10 sections, 14 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasi-ergodic theorems associated to Feynman--Kac transforms
  4. First computations for the expectation and variance
  5. Pointwise in time convergence results
  6. Quasi-ergodic limits for additive functionals
  7. Conditional functional weak law of large numbers
  8. Large deviation for additive functionals
  9. Examples
  10. Appendix

Key Result

Theorem 2.3

(KKT:2016) There is a unique minimizing function (called a ground state) $\phi_0:=\phi_0^{\mu, F}$ in specbot, that is, there exists $\phi_0 \in \mathcal{D}(\mathcal{E})$ such that $\int_E\phi_0^2(x)\mathfrak{m}({{\rm d}}x)=1$ and $\lambda_0=\mathcal{E}^{\mu, F}\left(\phi_0, \phi_0\right).$

Theorems & Definitions (29)

  • Remark 2.1
  • Example 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 19 more