Large Deviation Principle Research Articles

A Selection Principle for Weak KAM Solutions via Freidlin–Wentzell Large Deviation Principle of Invariant Measures

A Selection Principle for Weak KAM Solutions via Freidlin–Wentzell Large Deviation Principle of Invariant Measures

A large deviation principle for reflected SPDEs on infinite spatial domain

In this paper, we study a large deviation principle for a reflected stochastic partial differential equation on infinite spatial domain. A new sufficient condition for the weak convergence criterion proposed by Matoussi, Sabbagh and Zhang [A. Matoussi, W. Sabbagh and T.-S. Zhang, Large deviation principles of obstacle problems for quasilinear stochastic PDEs, Appl. Math. Optim. 83(2) (2021) 849–879] plays an important role in the proof.

Diffusion limited aggregation, resetting and large deviations of Brownian motion

Models of fractal growth commonly consider particles diffusing in a medium and that stick irreversibly to the forming aggregate when making contact for the first time. As shown by the well-known diffusion limited aggregation (DLA) model and its generalisations, the fractal dimension is sensitive to the nature of the stochastic motion of the particles. Here, we study the structures formed by finite-lived Brownian particles, i.e. particles constrained to find the aggregate within a prescribed time, and which are removed otherwise. This motion can be modelled by diffusion with stochastic resetting, a class of processes which has been widely studied in recent years. In the short lifetime limit, a very small fraction of the particles manage to reach the aggregate. Hence, growth is controlled by atypical Brownian trajectories, that move nearly in straight line according to a large deviation principle. In d dimensions, the resulting fractal dimension of the aggregate decreases from the DLA value and tends to 1, instead of increasing to d as expected from ballistic aggregation. In the zero lifetime limit one recovers the non-trivial model of ‘aggregation by the tips’ proposed long ago by R. Jullien [J. Phys. A: Math. Gen. 19, 2129 (1986)].

A large-deviation principle for birth–death processes with a linear rate of downward jumps

AbstractBirth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

Sharp moderate and large deviations for sample quantiles

In this article, we discuss the sharp moderate and large deviations between the quantiles of population and the quantiles of samples. Cramér type moderate deviations and Bahadur-Rao type large deviations are established with some mild conditions. The results refine the moderate and large deviation principles of Xu and Miao (2011).

Large Deviation Principle and Thermodynamic Limit of Chemical Master Equation via Nonlinear Semigroup

Large Deviation Principle and Thermodynamic Limit of Chemical Master Equation via Nonlinear Semigroup

Unimodal Maps Perturbed by Heteroscedastic Noise: An Application to a Financial Systems

We investigate and prove the mathematical properties of a general class of one-dimensional unimodal smooth maps perturbed with a heteroscedastic noise. Specifically, we investigate the stability of the associated Markov chain, show the weak convergence of the unique stationary measure to the invariant measure of the map, and show that the average Lyapunov exponent depends continuously on the Markov chain parameters. Representing the Markov chain in terms of random transformation enables us to state and prove the Central Limit Theorem, the large deviation principle, and the Berry-Esséen inequality. We perform a multifractal analysis for the invariant and the stationary measures, and we prove Gumbel’s law for the Markov chain with an extreme index equal to 1. In addition, we present an example linked to the financial concept of systemic risk and leverage cycle, and we use the model to investigate the finite sample properties of our asymptotic results

Large deviation principle for geometric and topological functionals and associated point processes

Large deviation principle for geometric and topological functionals and associated point processes

Gaussian Fluctuations for Interacting Particle Systems with Singular Kernels

We consider the asymptotic behaviour of the fluctuations for the empirical measures of interacting particle systems with singular kernels. We prove that the sequence of fluctuation processes converges in distribution to a generalized Ornstein–Uhlenbeck process. Our result considerably extends classical results to singular kernels, including the Biot–Savart law. The result applies to the point vortex model approximating the 2D incompressible Navier–Stokes equation and the 2D Euler equation. We also obtain Gaussianity and optimal regularity of the limiting Ornstein–Uhlenbeck process. The method relies on the martingale approach and the Donsker–Varadhan variational formula, which transfers the uniform estimate to some exponential integrals. Estimation of those exponential integrals follows by cancellations and combinatorics techniques and is of the type of the large deviation principle.

Large deviations for the two-time-scale stochastic convective Brinkman-Forchheimer equations

The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations.

Large deviation principle for the stochastic Cahn-Hilliard/Allen-Cahn equation with fractional noise

In this work, we consider the stochastic Cahn-Hilliard/Allen-Cahn equation with fractional noise, which is fractional in time and white in space. We obtain the existence, uniqueness, and Hölder regularity of the solution. Also, using a weak convergence approach, we prove that the law of solution associated with the above equation with a small perturbation satisfies the large deviation principle in the Hölder norm.

Uniform large deviations of fractional stochastic equations with polynomial drift on unbounded domains

In this paper, we first prove a uniform contraction principle for verifying the uniform large deviation principles of locally Hölder continuous maps in Banach spaces. We then show the local Hölder continuity of the solutions of a class of fractional parabolic equations with polynomial drift of any order defined on [Formula: see text]. We finally establish the large deviation principle of the fractional stochastic equations uniformly with respect to bounded initial data, despite the solution operators are not compact due to the non-compactness of Sobolev embeddings on unbounded domains.

Large deviations of fractional stochastic equations with non-Lipschitz drift and multiplicative noise on unbounded domains

This paper is concerned with the large deviation principle of the non-local fractional stochastic reaction-diffusion equation with a polynomial drift of arbitrary degree driven by multiplicative noise defined on unbounded domains. We first prove the strong convergence of the solutions of a control equation with respect to the weak topology of controls, and then show the convergence in distribution of the solutions of the stochastic equation when the noise intensity approaches zero. We finally establish the large deviations of the stochastic equation by the weak convergence method. The main difficulty of the paper is caused by the non-compactness of Sobolev embeddings on unbounded domains, and the idea of uniform tail-ends estimates is employed to circumvent the obstacle in order to obtain the tightness of distribution laws of the stochastic equation and the precompactness of the control equation.

On multifractal analysis and large deviations of singular hyperbolic attractors

In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class whose periodic orbits are all homoclinically related and such that the space of ergodic probability measures is connected, we prove that: (i) level sets associated to continuous observables are dense in the homoclinic class and satisfy a variational principle; (ii) irregular sets are either empty or are Baire generic and carry full topological entropy. The assumptions are satisfied by C 1-generic singular hyperbolic attractors and Cr -generic geometric Lorenz attractors . Finally we prove level-2 large deviations bounds for weak Gibbs measures, which provide a large deviations principle in the special case of Gibbs measures. The main technique we apply is the horseshoe approximation property.

Asymptotics of rectangular spherical integrals

In this article we study the Dyson Bessel process, which describes the evolution of singular values of rectangular matrix Brownian motions, and prove a large deviation principle for its empirical particle density. We then use it to obtain the asymptotics of the so-called rectangular spherical integrals as m,n go to infinity while m/n converges to a positive constant. We also improve and give a new proof of the asymptotics for the Harish-Chandra-Itzykson-Zuber integral derived in [37,38].

LARGE DEVIATIONS FOR STOCHASTIC VOLTERRA EQUATIONS WITH REFLECTION IN HÖLDERIAN NORM

In this paper, we study the large deviations principle \textbf{(LDP)} of the Volterra process with reflection in H\"olderian norm by using the Azencott method. As an application, we obtain the large deviations principle \textbf{(LDP)} of a perturbed reflected diffusion process driven by the Fractional Brownian Motion with Hurst parameter $H \in [\frac{1}{2},1)$.

Maximal large deviations and slow recurrences in weakly chaotic systems

We prove a maximal-type large deviation principle for dynamical systems with arbitrarily slow polynomial mixing rates. Also several applications, particularly to billiard systems, are presented.

On the small noise limit in the Smoluchowski-Kramers approximation of nonlinear wave equations with variable friction

We study the validity of a large deviation principle for a class of stochastic nonlinear damped wave equations, including equations of Klein-Gordon type, in the joint small mass and small noise limit. The friction term is assumed to be state dependent. We also provide the proof of the Smolchowski-Kramers approximation for the case of variable friction, non-Lipschitz nonlinear term and unbounded diffusion.

A sample-path large deviation principle for dynamic Erdős–Rényi random graphs

A sample-path large deviation principle for dynamic Erdős–Rényi random graphs

Large deviation principle for a stochastic process with random reinforced relocations

Stochastic processes with random reinforced relocations have been introduced in a series of papers by Boyer and co-authors (Boyer and Solis Salas 2014, Boyer and Pineda 2016, Boyer, Evans and Majumdar 2017) to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some ‘memory kernel’, and jumps to its value at that random time. We prove a quenched large deviation principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markovian due to relocations. Furthermore, the random inter-relocation times act as a random environment.