Large Deviation Principle Research Articles

Moderate deviations of generalized N-urn Ehrenfest models

This paper is a further investigation of the generalized N-urn Ehrenfest model introduced in Xue [Hydrodynamics of the generalized N-urn Ehrenfest model, Potential Anal. 2022. Online. Available at https://rdcu.be/cFzrj]. A moderate deviations principle from the hydrodynamic limit of the model is derived. The proof of this main result follows an exponential martingale approach introduced in C. Kipnis et al. [Hydrodynamics and large deviation for simple exclusion processes, Commun. Pure Appl. Math. 42 (1989), pp. 115–137], where a replacement lemma plays the key role. To prove the replacement lemma, the large deviations principle of the model given in Xue [Hydrodynamics of the generalized N-urn Ehrenfest model, Potential Anal. 2022. Online. Available at https://rdcu.be/cFzrj] is utilized. As applications of our main result, we further give moderate deviations of hitting times of our process and moderate deviations of our process at given moments.

Large Deviation Principle for Random Permutations

Abstract We derive a large deviation principle for random permutations induced by probability measures of the unit square, called permutons. These permutations are called $\mu $-random permutations. We also introduce and study a new general class of models of random permutations, called Gibbs permutation models, which combines and generalizes $\mu $-random permutations and the celebrated Mallows model for permutations. Most of our results hold in the general setting of Gibbs permutation models. We apply the tools that we develop to the case of $\mu $-random permutations conditioned to have an atypical proportion of patterns. Several results are made more concrete in the specific case of inversions. For instance, we prove the existence of at least one phase transition for a generalized version of the Mallows model where the base measure is non-uniform. This is in contrast with the results of Starr (2009, 2018) on the (standard) Mallows model, where the absence of phase transition, that is, phase uniqueness, was proven. Our results naturally lead us to investigate a new notion of permutons, called conditionally constant permutons, which generalizes both pattern-avoiding and pattern-packing permutons. We describe some properties of conditionally constant permutons with respect to inversions. The study of conditionally constant permutons for general patterns seems to be a new challenging problem.

Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models

We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP does not require strict self-similarity assumptions on the Volterra process. For this reason, we are able to apply such an LDP to two notable examples of non-self-similar rough volatility models: models where the volatility is given as a function of a log-modulated fractional Brownian motion (Bayer, C., F. Harang, and P. Pigato. 2021. “Log-Modulated Rough Stochastic Volatility Models.” SIAM Journal on Financial Mathematics 12 (3): 1257–1284), and models where it is given as a function of a fractional Ornstein–Uhlenbeck (fOU) process (Gatheral, J., T. Jaisson, and M. Rosenbaum. 2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949). In both cases, we derive consequences for short-maturity European option prices implied volatility surfaces and implied volatility skew. In the fOU case, we also discuss moderate deviations pricing and simulation results.

Large deviations of multichordal $\operatorname{SLE}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians

We prove a strong large deviation principle (LDP) for multiple chordal \operatorname{SLE}_{0+} curves with respect to the Hausdorff metric. In the single-chord case, this result strengthens an earlier partial result by the second author. We also introduce a Loewner potential, which in the smooth case has a simple expression in terms of zeta-regularized determinants of Laplacians. This potential differs from the LDP rate function by an additive constant depending only on the boundary data, which satisfies PDEs arising as a semiclassical limit of the Belavin–Polyakov–Zamolodchikov equations of level 2 in conformal field theory with central charge c \to -\infty . Furthermore, we show that every multichord minimizing the potential in the upper half-plane for given boundary data is the real locus of a rational function and is unique, thus coinciding with the \kappa \to 0+ limit of the multiple \operatorname{SLE}_\kappa . As a by-product, we provide an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition with a Möbius transformation.

Unified Limits and Large Deviation Principles for β-Laguerre Ensembles in Global Regime

Unified Limits and Large Deviation Principles for β-Laguerre Ensembles in Global Regime

Reflecting image-dependent SDEs in Wasserstein space and large deviation principle

In this article, we study a class of reflecting stochastic differential equations whose coefficients depend on image measures of solutions under a given initial measure in Wasserstein space . By the penalization method, the image process, which is a diffusion process in , is constrained in a priori given domain . The large deviation principle for this reflecting image process is also established by weak convergence method.

Small noise asymptotics of multi-scale McKean-Vlasov stochastic dynamical systems

The main aim of this work is to investigate small noise limiting behavior of multi-scale McKean-Vlasov stochastic dynamical systems, where we allow the coefficients depend on the distributions of both slow and fast components. Firstly, the strong convergence in the functional law of large numbers is established by the time discretization scheme. Secondly, in order to characterize the probability of deviations away from the averaged limit, we prove the large deviation principle by the weak convergence approach for McKean-Vlasov equations.

Large deviations in the early Universe

Fluctuations play a critical role in cosmology. They are relevant across a range of phenomena from the dynamics of inflation to the formation of structure. In many cases, these fluctuations are coarse grained and follow a Gaussian distribution as a consequence of the Central Limit Theorem. Yet, some classes of observables are dominated by rare fluctuations and are sensitive to the details of the underlying microphysics. In this paper, we argue that the Large Deviation Principle can be used to diagnose when one must to appeal to the fundamental description. Concretely, we investigate the regime of validity for the Fokker-Planck equation that governs Stochastic Inflation. For typical fluctuations, this framework leads to the central limit-type behavior expected of a random walk. However, fluctuations in the regime of the Large Deviation Principle are determined by instanton-like saddle points accompanied by a new energy scale. When this energy scale is above the UV cutoff of the EFT, the tail is only calculable in the microscopic description. We explicitly demonstrate this phenomenon in the context of determining the phase transition to eternal inflation, the distribution of scalar field fluctuations in de Sitter, and the production of primordial black holes.

A small time large deviation principle for stochastic differential delay equations

In this paper, we prove a small time large deviation principle for a type of stochastic differential delay equations. The key point is to translate small time asymptotic behavior problems into Freidlin–Wentzell type large deviation problems. The weak convergence method plays an important role in obtaining the large deviation principle.

Large deviations of [formula omitted]-blocks of regularly varying time series and applications to cluster inference

In the regularly varying time series setting, a cluster of exceedances is a short period for which the supremum norm exceeds a high threshold. We propose to study a generalization of this notion considering short periods, or blocks, with ℓp−norm above a high threshold. Our main result derives new large deviation principles of extremal ℓp−blocks, which guide us to define and characterize spectral cluster processes in ℓp. We then study cluster inference in ℓp to motivate our results. We design consistent disjoint blocks methods to infer features of cluster processes. Our inferential setting promotes the use of large empirical quantiles from the ℓp−norm of blocks as threshold levels which eases implementation and also facilitates comparison for different p>0. Our approach highlights the advantages of cluster inference based on extremal ℓα−blocks, where α>0 is the index of regular variation of the series. We focus on inference of important indices in extreme value theory, e.g., the extremal index.

Large deviations principle via Malliavin calculus for the Navier–Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker–Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier–Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in [17] in a discrete-time setting and extended in [26] to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman–Kac semigroup, which we verify by using Malliavin calculus.

Large deviation for a 3D globally modified Cahn–Hilliard–Navier–Stokes model under random influences

In this article, we derive a large deviation principle of the strong solution of the 3D globally modified Cahn–Hilliard–Navier–Stokes model under random influences. The model consists of the stochastic globally modified Navier–Stokes equations for the velocity, coupled with a Cahn–Hilliard equation for the order (phase) parameter. The proof relies on the weak convergence method that was introduced in Budhiraja et al. (2011) and based on a variational representation on infinite-dimensional Brownian motion.

Precise large deviations in a non stationary risk model with arbitrary dependence between subexponential claim sizes and waiting times

Consider a risk model in which the claims follow a non stationary arrival process that satisfies a large deviation principle. Supposing that the claim sizes form a sequence of i.i.d. random variables with subexponential tail, precise large deviations for the aggregate claims are obtained, by allowing the claim-sizes and claim inter-arrival (waiting) times to be arbitrarily dependent.

АСИМПТОТИЧЕСКИЕ МЕТОДЫ В ЗАДАЧАХ СУПЕРВИЗОРНОГО УПРАВЛЕНИЯ АВТОНОМНЫХ ПОДВОДНЫХ РОБОТОВ

The work is devoted to the relevant problem of control synthesis for autonomous underwater vehicles(AUVs). Since AUVs must perform actions in accordance with a given program under conditionsof a volatile predictable environment, it is necessary to provide working tools, the use of which, togetherwith the AUV position and orientation control requires, along with continuous local stabilization means,implementation of supervising and coordinated control algorithms at the upper level. Such a two-levelcontrol scheme, which can be called supervisory, is proposed in this paper to be implemented using twotypes of asymptotic methods: to separate movements into fast and slow, an apparatus for analyzing singularly perturbed differential equations is used, and the upper level control is based on the principleof large deviations. The general synthesis task is to control slow movements and stabilize fast movements.At the same time, in the stochastic problem formulation, it is assumed that there is a randomperturbation in fast movements. Given that fast movements are stabilized, it is high probable that thenoise influence averaged and does not significantly affect the behavior of slow variables. However, withsufficiently long observation it is possible to find a situation where at some period the perturbation valuesare not only uncompensated, but, on the contrary, line up in a sequence as specially intended toform an apparent deviation of the slow sub-vector from equilibrium. It is known from the theory of largedeviations that such a trajectory is the only and most likely of all those leading to a certain critical event.At the same time, the phase of this process can be used to judge the critical event proximity. For thisreason, the large deviations theory usage makes it possible to organize the control of deviations from aiven object’ t jecto y, capable of giving estimates on the probability of the controlled deviationscritical values. As a result, it is shown that if accelerations are formed by fast subsystems, then in slowsubsystems it is possible not only to achieve acceptable quality and accuracy at a fixed interval, but alsoto ensure this regardless of the disturbances. The validity of the proposed approach to synthesis, basedon the separation of movements and the large deviations theory, is shown on the example of an autonomousunderwater vehicle with two rudders (fore and aft) in the task of controlling longitudinal motion ata given depth. Simulation results and their discussion are presented.

A large deviation principle for fluids of third grade

ABSTRACT This article establishes a large deviation principle for a non-Newtonian fluid of differential type, filling a two-dimensional non-axisymmetric bounded domain with slip boundary conditions. More precisely, we show that the solutions of small stochastic white noise perturbations of the third grade fluid equations converges to the deterministic solution, as the intensity of the noise goes to zero. Moreover, this convergence has an exponential rate given by a suitable rate function. To establish such asymptotic result, we follow the weak convergence approach introduced by Budhiraja, Dupuis and Ellis.

Stochastic Observability and Convergent Analog State Estimation of Randomly Switched Linear Systems With Unobservable Subsystems

This article investigates observability and observer design for randomly switched linear systems (RSLSs) whose subsystems are all unobservable. Conditions for determining the analog state uniquely during operation, defined as stochastic observability, are studied. This article establishes probabilistic descriptions of stochastic observability for fast switching RSLSs. Design methods for subsystem observers and their organization for estimating the entire state are introduced. Convergence properties are established, including strong convergence and exponential convergence rate. Estimation error probabilities under finite data are derived by using the large deviation principles. Some critical structural conditions are characterized that permit organization of subsystem observers for achieving a convergent observer for the entire state. Examples and simulation case studies are presented to illustrate the main results of this article.

Large deviations for quadratic functionals of stable Gauss–Markov chains and entropy production

In this paper, we establish a large deviation principle for the entropy production rate of possible non-stationary, centered stable Gauss–Markov chains, verifying the Gallavotti–Cohen symmetry. We reach this goal by developing a large deviation theory for quasi-Toeplitz quadratic functionals of multivariate centered stable Gauss–Markov chains, which differ from a perfect Toeplitz form by the addition of quadratic boundary terms.

Large deviation principle for the intersection measure of Brownian motions on unbounded domains

Nous considérons la mesure d’intersection ℓtIS de p mouvements browniens indépendants sur Rd. Nous prouvons un principe de grande déviation pour la mesure d’intersection normalisée t−pℓtIS lorsque t tend vers l’infini, avant de sortir d’un domaine D⊂Rd (qui peut être non borné) avec une frontière lisse. Ce travail généralise [Comm. Pure Appl. Math. 66 (2013) 263–306] dans lequel D est borné. La contribution essentielle de cet article est de prouver, par une application de la relation de Chapman–Kolmogorov, une estimation sur-exponentielle pour la mesure d’intersection des mouvements browniens tués sur un tel D. Ce nouvel argument apporte aussi une preuve plus simple dans le cas des domaines bornés.

Large Deviation Principle of Nonconventional Ergodic Averages

Large Deviation Principle of Nonconventional Ergodic Averages

A large-deviations principle for all the components in a sparse inhomogeneous random graph

We study an inhomogeneous sparse random graph, {mathcal G }_N, on [N]={1,dots ,N} as introduced in a seminal paper by Bollobás et al. (Random Struct Algorithms 31(1):3–122, 2007): vertices have a type (here in a compact metric space {mathcal S }), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit Nrightarrow infty , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size asymp N). In doing so, we derive explicit logarithmic asymptotics for the probability that {mathcal G }_N is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of {mathcal G }_N. In particular, we recover the criterion for the existence of the phase transition given in Bollobás et al. (2007).