Large deviation principles for generalized Feynman-Kac functionals and its applications

Kim, Daehong; Kuwae, Kazuhiro; Tawara, Yoshihiro

doi:10.2748/tmj/1466172769

Cited by 9 publications

(4 citation statements)

References 36 publications

(29 reference statements)

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“…It is however present in the initial approach by Donsker and Varadhan. To our knowledge, such approach is only explicitly described in [23], in a few derived papers as in [21], and in [17]. As in our approach, the latter extend by this way the alternative approach for the convergence to QSDs that they have just obtained.…”

Section: Introductionmentioning

confidence: 79%

“…As in our approach, the latter extend by this way the alternative approach for the convergence to QSDs that they have just obtained. Especially in the case of continuous-time and continuous-space processes, spectral methods are most commonly exploited and lead to the study of Dirichlet forms, as notably in [23] or [21]. In the case of discrete-time processes, perturbation theories have also been been introduced to look at such large deviations [19].…”

Section: Introductionmentioning

confidence: 99%

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Large deviations for surviving trajectories of general Markov processes

Velleret

2020

Preprint

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The purpose of this paper is to ensure the conditions of Gärtner-Ellis Theorem for evaluations of the empirical measure. We show that up-to-date conditions for ensuring the convergence to a quasi-stationary distribution can be applied efficiently. By this mean, we are able to prove Large Deviation results even with a conditioning that the process is not extinct at the end of the evaluation. The domain on which these Large Deviation results apply is implicitly given by the range of penalization for which one can prove the above-mentioned results of quasi-stationarity. We propose a way to relate the range of controlled deviations to the range of admissible penalization. Central Limit Theorems are deduced from these results of Large Deviations. As an application, we consider the empirical measure of position and jumps of a continuous-time process on a unbounded domain of R d . This model is inspired by the adaptation of a population to a changing environment. Jumps makes it possible for the process to face a deterministic dynamics leading to high extinction areas.

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Section: Introductionmentioning

confidence: 79%

Section: Introductionmentioning

confidence: 99%

Large deviations for surviving trajectories of general Markov processes

Velleret

2020

Preprint

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“…excessive) functions belonging to F loc . Let N Rαν t be the CAF locally of zero energy appeared in the generalized Fukushima decomposition (see[31, Theorem 6.1]). The following lemma is a variant of[17, Lemma 5.4.1].…”

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confidence: 99%

Stability of estimates for fundamental solutions under Feynman-Kac perturbations for symmetric Markov processes

Kim¹,

Kim²,

Kuwae³

2022

Preprint

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In this paper, when a given symmetric Markov process X satisfies the stability of global heat kernel two-sided (upper) estimates by Markov perturbations (See Definition 1.2), we give a necessary and sufficient condition on the stability of global two-sided (upper) estimates for fundamental solution of Feynman-Kac semigroup of X. As a corollary, under the same assumptions, a weak type global two-sided (upper) estimates holds for the fundamental solution of Feynman-Kac semigroup with (extended) Kato class conditions for measures. This generalizes all known results on the stability of global integral kernel estimates by symmetric Feynman-Kac perturbations with Kato class conditions in the framework of symmetric Markov processes.

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“…Thus we cannot apply directly the methods exposed in [16,21] to our cases. Nevertheless, we can obtain our results by noting the recent developments of the generalized Feynman-Kac transforms and their related topics studied in [8, 12,13,14]. The remainder of the paper is organized as follows.…”

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confidence: 99%

Convergence of measures penalized by generalized Feynman-Kac transforms

Kim¹,

Matsuura²

2016

Tohoku Math. J. (2)

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We prove the existence of limiting laws for symmetric stable-like processes penalized by generalized Feynman-Kac functionals and characterize them by the gauge functions and the ground states of Schrodinger type operators. Introduction.In this paper we study limit theorems for symmetric stable-like processes on R d penalized by normalized Feynman-Kac functionals as the weight processes.Penalizing measures by appropriate weight processes can be understood as a change-ofmeasure phenomenon and such modifications have been studied by many authors ([16, 18, 21, 25]). In penalizations, the weights play a role analogous to a Girsanov transform in which its martingale property allows to define a new probability measure, do not allow immediately to create a new weighted probability measure that may emerge in the limit of weight processes. When such a limit exists, it is called the penalized probability measure associated with the weights.In [18], Roynette, Vallois and Yor have studied limit theorems for Wiener processes penalized by various weight processes. In [25], the authors studied limit theorems for the onedimensional symmetric stable process penalized by Feynman-Kac transforms with negative (killing) additive functionals, and they called their limit theorems the Feynman-Kac penalizations. It turns out that their methods are not available in multi-dimensional cases. In [21], Takeda extended their results to Feynman-Kac transforms with positive (creation) continuous additive functionals corresponding to positive smooth measures for multi-dimensional symmetric stable processes on R d by classifying associated Schrödinger operators of Feynman-Kac semigroups into the subcritical, critical and supercritical cases, and by characterizing the penalized measures in each cases. Recently, this result was extended to non-local Feynman-Kac transforms by Matsuura [16].The purpose of this paper is to extend the previous results for penalization problems to the so-called generalized Feynman-Kac transforms. More precisely, let X = (Ω, F ∞ , F t ,X t ,P x ) be the symmetric α-stable-like process on R d with 0 < α < 2 and (E, F ) the associated

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Large deviation principles for generalized Feynman-Kac functionals and its applications

Cited by 9 publications

References 36 publications

Large deviations for surviving trajectories of general Markov processes

Large deviations for surviving trajectories of general Markov processes

Stability of estimates for fundamental solutions under Feynman-Kac perturbations for symmetric Markov processes

Convergence of measures penalized by generalized Feynman-Kac transforms

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