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We study the small deviation problem $\log\mathbb{P}(\sup_{t\in[0,1]}|X_t|\leq\varepsilon)$, as $\varepsilon\to0$, for general L\'{e}vy processes $X$. The techniques enable us to determine the asymptotic rate for general real-valued…

Probability · Mathematics 2009-09-25 Frank Aurzada , Steffen Dereich

Let $X$ be a L\'evy process with regularly varying L\'evy measure $\nu$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed…

Probability · Mathematics 2017-12-12 Chang-Han Rhee , Jose Blanchet , Bert Zwart

The aim of this paper is to investigate the large deviations for a class of slow-fast mean-field diffusions, which extends some existing results to the case where the laws of fast process are also involved in the slow component. Due to the…

Probability · Mathematics 2026-04-28 Wei Hong , Wei Liu , Shiyuan Yang

We investigate the Large Deviation behavior in small time of continuous Gaussian processes. We introduce a general procedure allowing to derive Large Deviation Principles in small time starting from the well understood context of Large…

Probability · Mathematics 2023-01-11 Paolo Baldi , Barbara Pacchiarotti

We present a new description of the known large deviation function of the classical symmetric simple exclusion process by exploiting its connection with the quantum symmetric simple exclusion processes and using tools from free probability.…

Mathematical Physics · Physics 2023-09-27 Michel Bauer , Denis Bernard , Philippe Biane , Ludwig Hruza

We study the large deviation estimates for the short time asymptotic behavior of a strongly degenerate diffusion process. Assuming a nilpotent structure of the Lie algebra generated by the driving vector fields, we obtain a graded large…

Probability · Mathematics 2019-01-30 Gérard Ben Arous , Jing Wang

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in $n$ fixed points at $n$ fixed past instants. In particular, functional large deviation results are stated for small time. Several…

Probability · Mathematics 2016-04-06 L. Caramellino , B. Pacchiarotti

The main results in this paper concern large deviations for families of non-Gaussian processes obtained as suitable perturbations of continuous centered multivariate Gaussian processes which satisfy a large deviation principle. We present…

Probability · Mathematics 2023-07-06 C. Macci , B. Pacchiarotti

The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as…

Statistical Mechanics · Physics 2009-08-20 Hugo Touchette

We study large deviations asymptotics for a class of unbounded additive functionals, interpreted as normalized accumulated areas, of one-dimensional Langevin diffusions with sub-linear gradient drifts. Our results provide parametric…

Probability · Mathematics 2023-10-23 Mihail Bazhba , Jose Blanchet , Roger J. A. Laeven , Bert Zwart

We present here a simple method for computing the large deviation of long time average for stochastic jump processes. We show that the computation of the rate function can be reduced to that of a partial differential equation governing the…

Statistical Mechanics · Physics 2020-04-22 Bahram Houchmandzadeh

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a…

Statistics Theory · Mathematics 2009-10-25 Yaming Yu

Fluctuation theorem is one of the major achievements in the field of nonequilibrium statistical mechanics during the past two decades. Steady-state fluctuation theorem of sample entropy production rate in terms of large deviation principle…

Probability · Mathematics 2016-08-24 Yong Chen , Hao Ge , Jie Xiong , Lihu Xu

In this paper we consider the persistence properties of random processes in Brownian scenery, which are examples of non-Markovian and non-Gaussian processes. More precisely we study the asymptotic behaviour for large $T$, of the probability…

Probability · Mathematics 2015-02-25 Fabienne Castell , Nadine Guillotin-Plantard , Frederique Watbled

We study the asymptotic behaviour of solutions of Forward Backward Stochastic Differential Equations in the coupled case, when the diffusion coefficient of the forward equation is multiplicatively perturbed by a small parameter that…

Probability · Mathematics 2013-02-27 Ana Bela Cruzeiro , André de Oliveira Gomes

We use a semi-Markov process method to calculate large deviations of counting statistics for three open quantum systems, including a resonant two-level system and resonant three-level systems in the $\Lambda$- and $V$-configurations. In the…

Statistical Mechanics · Physics 2023-12-15 Fei Liu

We obtain large deviation results for a two time-scale model of jump-diffusion processes. The processes on the two time scales are fully inter-dependent, the slow process has small perturbative noise and the fast process is ergodic. Our…

Probability · Mathematics 2016-09-19 Rohini Kumar , Lea Popovic

We extend the study by Ornstein and Weiss on the asymptotic behavior of the normalized version of recurrence times and establish the large deviation property for a certain class of mixing processes. Further, an estimator for entropy based…

Information Theory · Computer Science 2013-05-21 Siddharth Jain , Rakesh Kumar Bansal

Consider the normalized partial sums of a real-valued function $F$ of a Markov chain, \[\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.\] The chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $\mathsf {X}$, with…

Probability · Mathematics 2007-05-23 Sean P. Meyn

We establish a large deviation principle for the trajectories of Wiener processes subject to random resets to the origin occurring according to a Poisson process. In addition to the pathwise large deviation principle, we identify the rate…

Probability · Mathematics 2025-12-09 A. V. Logachov , O. M. Logachova , A. A. Yambartsev , K. A. Zaykov