Large Deviation Principle Research Articles

Large deviation principle of stochastic evolution equations with reflection

In this paper, we establish a large deviation principle for stochastic evolution equations with reflection in an infinite-dimensional ball. Weak convergence approach plays an important role.

Large Deviation Principle for Backward Stochastic Differential Equations with a Stochastic Lipschitz Condition on z

Large Deviation Principle for Backward Stochastic Differential Equations with a Stochastic Lipschitz Condition on <i>z</i>

Large Deviations for Small Noise Hypoelliptic Diffusion Bridges on Sub-Riemannian Manifolds

In this paper we study a large deviation principle of Freidlin–Wentzell type for pinned hypoelliptic diffusion measures associated with a natural sub-Laplacian on a compact sub-Riemannian manifold. To prove this large deviation principle, we use rough path theory and manifold-valued Malliavin calculus.

Monge-Ampère gravity: From the large deviation principle to cosmological simulations through optimal transport

Monge-Ampère gravity: From the large deviation principle to cosmological simulations through optimal transport

Symmetry Shapes Thermodynamics of Macroscopic Quantum Systems.

We derive a systematic approach to the thermodynamics of quantum systems based on the underlying symmetry groups. We first show that the entropy of a system can be described in terms of group-theoretical quantities that are largely independent of the details of its density matrix. We then apply our technique to generic N identical interacting d-level quantum systems. Using permutation invariance, we find that, for large N, the entropy displays a universal asymptotic behavior in terms of a function s(x) that is completely independent of the microscopic details of the model, but depends only on the size of the irreducible representations of the permutation group S_{N}. In turn, the equilibrium state of the system and macroscopic fluctuations around it are shown to satisfy a large deviation principle with a rate function f(x)=e(x)-β^{-1}s(x), where e(x) only depends on the ground state energy of particular subspaces determined by group representation theory, and β is the inverse temperature. We apply our theory to the transverse-field Curie-Weiss model, a minimal model of phase transition exhibiting an interplay of thermal and quantum fluctuations.

Wiener densities for the Airy line ensemble

AbstractThe parabolic Airy line ensemble is a central limit object in the KPZ (Kardar–Parisi–Zhang) universality class and related areas. On any compact set , the law of the recentered ensemble has a density with respect to the law of independent Brownian motions. We show where is an explicit, tractable, non‐negative function of . We use this formula to show that is bounded above by a ‐dependent constant, give a sharp estimate on the size of the set where as , and prove a large deviation principle for . We also give density estimates that take into account the relative positions of the Airy lines, and prove sharp two‐point tail bounds that are stronger than those for Brownian motion. These estimates are a key input in the classification of geodesic networks in the directed landscape. The paper is essentially self‐contained, requiring only tail bounds on the Airy point process and the Brownian Gibbs property as inputs.

High moments of the SHE in the clustering regimes

We analyze the high moments of the Stochastic Heat Equation (SHE) via a transformation to the attractive Brownian Particles (BPs), which are Brownian motions interacting via pairwise attractive drift. In those scaling regimes where the particles tend to cluster, we prove a Large Deviation Principle (LDP) for the empirical measure of the attractive BPs. Under the delta(-like) initial condition, we characterize the unique minimizer of the rate function and relate the minimizer to the spacetime limit shapes of the Kardar–Parisi–Zhang (KPZ) equation in the upper tails. The results of this paper are used in the companion paper [75] to prove an n-point, upper-tail LDP for the KPZ equation and to characterize the corresponding spacetime limit shape.

Outage Performance of Satellite-Terrestrial Channels With Shadowed Rician Fading via Large Deviations Principle

Outage Performance of Satellite-Terrestrial Channels With Shadowed Rician Fading via Large Deviations Principle

Large deviations for slow–fast processes on connected complete Riemannian manifolds

We consider a class of slow–fast processes on a connected complete Riemannian manifold M. The limiting dynamics as the scale separation goes to ∞ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with Hamilton–Jacobi–Bellman (HJB) equation techniques. Our main innovation is solving the comparison principle for viscosity solutions for the HJB equation on M and the construction of a variational viscosity solution for the non-smooth Hamiltonian, which lies at the heart of deriving the action integral representation for the rate function.

Exact Calculation of the Probabilities of Rare Events in Cluster-Cluster Aggregation.

We develop an action formalism to calculate probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels and establish a pathwise large deviation principle with total mass being the rate. As an application, the rate function for the number of surviving particles as well as the optimal evolution trajectory are calculated exactly for the constant, sum, and product kernels. For the product kernel, we argue that the second derivative of the rate function has a discontinuity. The theoretical results agree with simulations tailored to the calculation of rare events.

Large deviation principle for pseudo-monotone evolutionary equation

. In this article, we explore Freidlin-Wentzell’s large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework.

Dynamical large deviations for long-range interacting inhomogeneous systems without collective effects.

We consider the long-term evolution of a spatially inhomogeneous long-range interacting N-body system. Placing ourselves in the dynamically hot limit, i.e., assuming that the system only weakly amplifies perturbations, we derive a large deviation principle for the system's empirical angle-averaged distribution function. This result extends the classical ensemble-averaged kinetic theory given by the so-called inhomogeneous Landau equation, as it specifies the probability of typical and large dynamical fluctuations. We detail the main properties of the associated large deviation Hamiltonian, particularly focusing on how it complies with the system's conservation laws and possesses a gradient structure.

Large deviations in statistics of the local time and occupation time for a run and tumble particle.

We investigate the statistics of the local time T=∫_{0}^{T}δ(x(t))dt that a run and tumble particle (RTP) x(t) in one dimension spends at the origin, with or without an external drift. By relating the local time to the number of times the RTP crosses the origin, we find that the local time distribution P(T) satisfies the large deviation principle P(T)∼e^{-TI(T/T)} in the large observation time limit T→∞. Remarkably, we find that in the presence of drift the rate function I(ρ) is nonanalytic: we interpret its singularity as a dynamical phase transition of first order. We then extend these results by studying the statistics of the amount of time R that the RTP spends inside a finite interval (i.e., the occupation time), with qualitatively similar results. In particular, this yields the long-time decay rate of the probability P(R=T) that the particle does not exit the interval up to time T. We find that the conditional end-point distribution exhibits an interesting change of behavior from unimodal to bimodal as a function of the size of the interval. To study the occupation time statistics, we extend the Donsker-Varadhan large-deviation formalism to the case of RTPs, for general dynamical observables and possibly in the presence of an external potential.

Large Deviations of Invariant Measures of Stochastic Reaction–Diffusion Equations on Unbounded Domains

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document}. Since the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not compact and the spectrum of the Laplace operator on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces.

Anomalous scalings of fluctuations of the area swept by a Brownian particle trapped in a [formula omitted] potential

We study the fluctuations of the area A=∫0Tx(t)dt under a one-dimensional Brownian motion x(t) in a trapping potential ∼|x|, at long times T→∞. We find that typical fluctuations of A follow a Gaussian distribution with a variance that grows linearly in time (at large T), as do all higher cumulants of the distribution. However, large deviations of A are not described by the “usual” scaling (i.e., the large deviations principle), and are instead described by two different anomalous scaling behaviors: Moderately-large deviations of A, obey the anomalous scaling PA;T∼e−T1/3fA/T2/3 while very large deviations behave as PA;T∼e−TΨA/T2. We find the associated rate functions f and Ψ exactly. Each of the two functions contains a singularity, which we interpret as dynamical phase transitions of the first and third order, respectively. We uncover the origin of these striking behaviors by characterizing the most likely scenario(s) for the system to reach a given atypical value of A. We extend our analysis by studying the absolute area B=∫0T|x(t)|dt and also by generalizing to higher spatial dimension, focusing on the particular case of three dimensions.

Large Deviations Principle for Stochastic Delay Differential Equations with Super-linearly Growing Coefficients

Large Deviations Principle for Stochastic Delay Differential Equations with Super-linearly Growing Coefficients

Large deviations for the Yule–Walker estimator of near critical autoregressive processes

The large deviation principle is established for the Yule–Walker estimator of the near critical order one autoregressive process. The rate function is identified explicitly. Our result shows that, at the exponential scale, one cannot distinguish between near critical and the critical Yule–Walker estimators.

Large deviations for stochastic predator–prey model with Lévy noise

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations.

Large deviations for regime-switching diffusions with infinite delay

Focusing on a class of regime-switching functional diffusion processes with infinite delay, a Freidlin–Wentzell type large deviations principle (LDP) is established by using an extended contraction principle and an exponential approximation argument under a local one-side Lipschitz condition. The result is new even for functional diffusion processes with infinite delay without regime-switching. Several interesting examples are given to illustrate our results.

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