Uniform Large Deviation Research Articles

The ergodicity and uniform large deviations for the 1D stochastic Landau-Lifshitz-Bloch equation

In this article, we consider two aspects asymptotic behaviors for the 1D stochastic Landau-Lifshitz-Bloch equation driven by additive noise including the unique ergodicity and the uniform large deviations. Using a deterministic argument relying on the exponential moment and exponential stability estimates, we first establish the unique ergodicity of invariant measure in H 1. Then, the uniform large deviations is established in C ( [ 0 , T ] ; H 1 ) by the uniform contraction principle.

Uniform Large Deviation Principle for the Solutions of Two-Dimensional Stochastic Navier–Stokes Equations in Vorticity Form

Uniform Large Deviation Principle for the Solutions of Two-Dimensional Stochastic Navier–Stokes Equations in Vorticity Form

Uniform Large Deviations for Stochastic Generalized Burgers-Huxley Equation Driven by Multiplicative Noise

Uniform Large Deviations for Stochastic Generalized Burgers-Huxley Equation Driven by Multiplicative Noise

Uniform large deviations for stochastic Burgers–Huxley equation driven by multiplicative noise

In this paper, we establish a Freidlin–Wentzell type large deviation principle uniformly with respect to initial condition in bounded subsets of an infinite dimensional Banach space for stochastic Burgers–Huxley equation driven by multiplicative small noise. The proof is based on the weak convergence approach obtained by Salins, Budhiraja, Dupuis (Trans. Amer. Math. Soc., 2019).

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.

Random sampling in reproducing kernel spaces with mixed norm

In this work we consider random sampling of signals in (in)finite-dimensional reproducing kernel spaces with mixed norm. Here the random sampling refers to randomly taken sampling positions according to some probability measure. We study the stability of random sampling procedure by establishing sampling inequality that holds with high probability when the sampling size is large. We establish the probabilistic sampling inequality though a combination of mathematical analysis and probabilistic analysis. The main tools we use are covering number of signal (function) space and (uniform) large deviation inequality for a sequence of random variables. We provide a concise proof and our proof leads to explicit and transparent estimates involved in the probability with which the sampling inequality holds.

Uniform large deviation principles of fractional stochastic reaction-diffusion equations on unbounded domains

This paper is concerned with uniform large deviation principles of fractional stochastic reaction-diffusion equations driven by additive noise defined on unbounded domains where the solution operator is non-compact and hence the result of [32] does not apply. The nonlinear drift is assumed to be locally Lipschitz continnous instead of being globally Lipschitz continuous. We first prove a large deviation principle for a fractional linear stochastic equation by the weak convergence method, and then show a uniform large deviation principle for the fractional nonlinear equation by a uniform contraction principle, despite the Sobolev embeddings are non-compact in unbounded domains. The result of the paper regrading the uniform large deviations can be applied to investigate the exit time and exit place of the solutions of the stochastic reaction-diffusion equations from a given domain in the phase space.

Uniform large deviations for a class of semilinear stochastic partial differential equations driven by a Brownian sheet

Uniform large deviations for a class of semilinear stochastic partial differential equations driven by a Brownian sheet

Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a central limit theorem and a large deviations principle.

Asymptotics of quasi-stationary distributions of small noise stochastic dynamical systems in unbounded domains

AbstractWe consider a collection of Markov chains that model the evolution of multitype biological populations. The state space of the chains is the positive orthant, and the boundary of the orthant is the absorbing state for the Markov chain and represents the extinction states of different population types. We are interested in the long-term behavior of the Markov chain away from extinction, under a small noise scaling. Under this scaling, the trajectory of the Markov process over any compact interval converges in distribution to the solution of an ordinary differential equation (ODE) evolving in the positive orthant. We study the asymptotic behavior of the quasi-stationary distributions (QSD) in this scaling regime. Our main result shows that, under conditions, the limit points of the QSD are supported on the union of interior attractors of the flow determined by the ODE. We also give lower bounds on expected extinction times which scale exponentially with the system size. Results of this type when the deterministic dynamical system obtained under the scaling limit is given by a discrete-time evolution equation and the dynamics are essentially in a compact space (namely, the one-step map is a bounded function) have been studied by Faure and Schreiber (2014). Our results extend these to a setting of an unbounded state space and continuous-time dynamics. The proofs rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of continuous-time dynamical systems.In general, QSD for Markov chains with absorbing states and unbounded state spaces may not exist. We study one basic family of binomial-Poisson models in the positive orthant where one can use Lyapunov function methods to establish existence of QSD and also to argue the tightness of the QSD of the scaled sequence of Markov chains. The results from the first part are then used to characterize the support of limit points of this sequence of QSD.

Systems of small-noise stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over all initial data

Large deviations principles characterize the exponential decay rates of the probabilities of rare events. Cerrai and Röckner (2004) proved that systems of stochastic reaction–diffusion equations satisfy a large deviations principle that is uniform over bounded sets of initial data.This paper proves uniform large deviations results for a system of stochastic reaction–diffusion equations in a more general setting than Cerrai and Röckner. Furthermore, this paper identifies two common situations where the large deviations principle is uniform over unbounded sets of initial data, enabling the characterization of Freidlin–Wentzell exit time and exit shape asymptotics from unbounded sets.

Uniform Large Deviations for a Class of Burgers-Type Stochastic Partial Differential Equations in any Space Dimension

Uniform Large Deviations for a Class of Burgers-Type Stochastic Partial Differential Equations in any Space Dimension

A non-exponential extension of Sanov’s theorem via convex duality

AbstractThis work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

Uniform large deviation principles for Banach space valued stochastic evolution equations

We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDEs) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform LDP over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite-dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak- ⋆ \star compactness of closed bounded sets in the double dual space. We prove that a modified version of our SDE satisfies a uniform Laplace principle over weak- ⋆ \star compact sets and consequently a uniform over bounded sets LDP. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite-dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and two-dimensional stochastic Navier–Stokes equations with multiplicative noise.

On Uniform Large Deviations Principle for Multi-valued SDEs via the Viscosity Solution Approach

This paper deals with the uniform large deviations for multivalued stochastic differential equations (MSDEs for short) by applying a stability result of the viscosity solutions of second order Hamilton-Jacobi-Belleman equations with multivalued operators. Moreover, the large deviation principle is uniform in time and in starting point.

Equivalences and counterexamples between several definitions of the uniform large deviations principle

This paper explores the equivalences between four definitions of uniform large deviations principles and uniform Laplace principles found in the literature. Counterexamples are presented to illustrate the differences between these definitions and specific conditions are described under which these definitions are equivalent to each other. A fifth definition called the equicontinuous uniform Laplace principle (EULP) is proposed and proven to be equivalent to Freidlin and Wentzell’s definition of a uniform large deviations principle. Sufficient conditions that imply a measurable function of infinite dimensional Wiener process satisfies an EULP using the variational methods of Budhiraja, Dupuis and Maroulas are presented. This theory is applied to prove that a family of Hilbert space valued stochastic equations exposed to multiplicative noise satisfy a uniform large deviations principle that is uniform over all initial conditions in bounded subsets of the Hilbert space, and under stronger assumptions is uniform over initial conditions in unbounded subsets too. This is an improvement over previous weak convergence methods which can only prove uniformity over compact sets.

Large deviations for neutral functional SDEs with jumps

In this paper, by the weak convergence method, based on a variational representation for positive functionals of a Poisson random measure and Brownian motion, we establish uniform large deviation principles (LDPs) for a class of neutral stochastic differential equations driven by jump processes. As a byproduct, we also obtain uniform LDPs for neutral stochastic differential delay equations which, in particular, allow the coefficients to be highly nonlinear with respect to the delay argument.

A tightness property of a symmetric Markov process and the uniform large deviation principle

Previously, we considered a large deviation for occupation measures of a symmetric Markov processes under the condition that its resolvent possesses a kind of tightness property. In this paper, we prove that if the Markov process is conservative, then the tightness property implies the uniform hyper-exponential recurrence, which leads us to the uniform large deviation principle.

Fluctuation relations without uniform large deviations

We study the fluctuations of a stochastic Maxwell–Lorentz particle model driven by an external field to determine the extent to which fluctuation relations are related to large deviations. Focusing on the total entropy production of this model in its steady state, we show that, although the probability density of this quantity globally satisfies (by definition) a fluctuation relation, its negative tail decays exponentially with time, whereas its positive tail decays slower than exponentially with time because of long collision-free trajectories. This provides an example of a physical system for which the fluctuation relation does not derive, as commonly thought, from a probability density decaying everywhere exponentially with time or, in other words, from a probability density having a uniform large deviation form.

A Strong Law for the Rate of Growth of Long Latency Periods in a Cloud Computing Service

Cloud-computing shares a common pool of resources across customers at a scale that is orders of magnitude larger than traditional multiuser systems. Constituent physical compute servers are allocated multiple ‘virtual machines' (VMs) to serve simultaneously. Each VM user should ideally be unaffected by others’ demand. Naturally, this environment produces new challenges for the service providers in meeting customer expectations while extracting an efficient utilization from server resources. We study a new cloud service metric that measures prolonged latency or delay suffered by customers. We model the workload process of a cloud server and analyze the process as the customer population grows. The capacity required to ensure that the average workload does not exceed a threshold over long segments is characterized. This can be used by cloud operators to provide service guarantees on avoiding long durations of latency. As part of the analysis, we provide a uniform large deviation principle for collections of random variables that is of independent interest.