We investigate the relation between the small deviation problem for a symmetric $α$-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric $α$-stable random processes, including unbounded Riemann--Liouville processes, weighted Riemann--Liouville processes and the ($d$-dimensional) $α$-stable sheet.
We offer a new point of view on the (Modified) Log-Sobolev inequality and lower bounds on the Ricci-curvature in the setting where the dynamics are obtained as the limit of Markov processes. In this setting, the large deviation rate function of the stationary measures of the Markov processes, plays the role of entropy. We define an entropy-information inequality (EII) that generalizes the (MLSI) and is equivalent to exponential decay of the rate function along the flow, and define an entropy-convexity inequality (ECI) that serves as an analogue of a lower bound on the Ricci-curvature in this s…
We consider random walk with bounded jumps on a hypercubic lattice of arbitrary dimension in a dynamic random environment. The environment is temporally independent and spatially translation invariant. We study the rate functions of the level-3 averaged and quenched large deviation principles from the point of view of the particle. In the averaged case the rate function is a specific relative entropy, while in the quenched case it is a Donsker-Varadhan type relative entropy for Markov processes. We relate these entropies to each other and seek to identify the minimizers of the level-3 to level…
We investigate the large deviation behaviour of a point process sequence based on a stationary symmetric stable non-Gaussian discrete-parameter random field using the framework of Hult and Samorodnitsky (2010). Depending on the ergodic theoretic and group theoretic structures of the underlying nonsingular group action, we observe different large deviation behaviours of this point process sequence. We use our results to study the large deviations of various functionals (e.g., partial sum, maxima, etc.) of stationary symmetric stable fields.
We obtain new transport-entropy inequalities and, as a by-product, new deviation estimates for the laws of two kinds of discrete stochastic approximation schemes. The first one refers to the law of an Euler like discretization scheme of a diffusion process at a fixed deterministic date and the second one concerns the law of a stochastic approximation algorithm at a given time-step. Our results notably improve and complete those obtained in [Frikha, Menozzi,2012]. The key point is to properly quantify the contribution of the diffusion term to the concentration regime. We also derive a general n…
We shall establish a large deviation principle for some occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. As a consequence, we obtain the exact rate of exponential ergodicity of the stochastic system under $τ$-topology.
We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on $\mathbb Z^d$ is in the domain of attraction of an operator-stable law on $\mathbb R^d$. Applications to random walks on wreath products (also known as lamplighter groups) are given.
This paper illustrates the power of the entropy method in addressing problems from large deviation theory. We provide and review entropy proofs for most fundamental results in large deviation theory, including Cramer's theorem, the Gärtner--Ellis theorem, and Sanov's theorem. Moreover, by the entropy method, we also strengthen Sanov's theorem to the strong version.
We establish a large deviation principle for the occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. The proof is based on a hyper-exponential recurrence criterion. Our result indicates a phenomenon that strong dissipation beats heavy tailed noises to produce a large deviation, it seems to us that this phenomenon has not been reported in the known literatures.
We study the object formally defined as γ\big([0,t]^{2}\big)=\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds-E\int\int_{[0,t]^{2}} | X_{s}- X_{r}|^{-σ} dr ds, where $X_{t}$ is the symmetric stable processes of index $0<β\le 2$ in $R^{d}$. When $β\leσ<\displaystyle\min \Big\{{3\over 2}β, d\Big\}$, this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for $γ$. This is applied to obtain results about stable processes in random potentials.
Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_T(x)= \int_0^T δ_x(X_s)ds$ the local time at the state $x$ and $ I_T= \sum\limits_{x\in\mathbb{Z}^d} l_T(x)^q $ the q-fold self-intersection local time (SILT). In \cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and Mörters obtain in \cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for the…
In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete…
Let $Ξ\subset\mathbb R^d$ be a set of centers chosen according to a Poisson point process in $\mathbb R^d$. Let $ψ$ be an allocation of $\mathbb R^d$ to $Ξ$ in the sense of the Gale-Shapley marriage problem, with the additional feature that every center $ξ\inΞ$ has an appetite given by a nonnegative random variable $α$. Generalizing some previous results, we study large deviations for the distance of a typical point $x\in\mathbb R^d$ to its center $ψ(x)\inΞ$, subject to some restrictions on the moments of $α$.
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 for diffusion processes have not been rigorously proved yet due to technical difficulties. Here we give a proof for the steady-state fluctuation theorem of a diffusion process in magnetic fields, with explicit expressions of the free energy function and rate function. The proof is based on the Karhunen-Loéve expansion of complex-valued Ornstein-Uhlenbeck pro…
We study the small deviation probabilities of a family of very smooth self-similar Gaussian processes. The canonical process from the family has the same scaling property as standard Brownian motion and plays an important role in the study of zeros of random polynomials. Our estimates are based on the entropy method, discovered in Kuelbs and Li (1992) and developed further in Li and Linde (1999), Gao (2004), and Aurzada et al. (2009). While there are several ways to obtain the result w.r.t. the $L_2$ norm, the main contribution of this paper concerns the result w.r.t. the supremum norm. In thi…
We give a short analytic proof of local large deviations for i.i.d. random variables in the domain of a multivariate $α$-stable law, $α\in(0,1)\cup(1,2]$. Our method simultaneously covers lattice and nonlattice distributions (and mixtures thereof), bypassing aperiodicity considerations. The proof applies also to the dynamical setting.
In this paper, we prove a process-level, also known as level-3 large deviation principle for a very general class of simple point processes, i.e. nonlinear Hawkes process, with a rate function given by the process-level entropy, which has an explicit formula.
We consider a stable but nearly unstable autoregressive process of any order. The bridge between stability and instability is expressed by a time-varying companion matrix $A_{n}$ with spectral radius $ρ(A_{n}) < 1$ satisfying $ρ(A_{n}) \rightarrow 1$. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of $A_{n}$ through $1-ρ(A_{n})$ and, as a by-product, we establish a moderate deviation principle for the OLS estimator when $Γ$, the renormalized asymptotic variance of the process, is invertible. Finally, when $Γ$ is singular…
Through this paper we analyze the ergodic properties of continuous time Markov chains with values on the one-dimensional spin lattice 1,...,d}^N (also known as the Bernoulli space). Initially, we consider as the infinitesimal generator the operator $L={\mc L}_A -I$, where \mc L_A is a discrete time Ruelle operator (transfer operator), and A:{1,...,d}^N \to R is a given fixed Lipschitz function. The associated continuous time stationary Markov chain will define the\emph{a priori}probability. Given a Lipschitz interaction V:\{1,...,d\}^{\bb N}\to \mathbb{R}, we are interested in Gibbs (equilibri…
Max stable laws are limit laws of linearly normalized partial maxima of indepen- dent, identically distributed (iid) random variables (rvs). These are analogous to stable laws which are limit laws of normalized partial sums of iid rvs. In this paper, we study entropy limit theorems for distribution functions in the max domain of attraction of max stable laws under linear normalization. More specifically, we study the problem of convergence of the Shannon entropy of linearly normalized partial maxima of iid rvs to the corresponding limit entropy when the linearly normalized partial maxima conve…
We study the motion of a particle in a random time-dependent vector field defined by the 2D Navier-Stokes system with a noise. Under suitable non-degeneracy hypotheses we prove that the empirical measures of the trajectories of the pair (velocity field, particle) satisfy the LDP with a good rate function. Moreover, we show that the law of a unique stationary solution restricted to the particle component possesses a positive smooth density with respect to the Lebesgue measure in any finite time. This allows one to define a natural concept of the entropy production, and to show that its time ave…
We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the …
We study a class of dissipative PDE's perturbed by an unbounded kick force. Under some natural assumptions, the restrictions of solutions to integer times form a homogeneous Markov process. Assuming that the noise is rough with respect to the space variables and has a non-degenerate law, we prove that the system in question satisfies a large deviation principle in tau-topology. Under some additional hypotheses, we establish a Gallavotti-Cohen type symmetry for the rate function of an entropy production functional and the strict positivity and finiteness of the mean entropy production in the st…
Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions $\mathscr L$ that induce a flow, given by $\mathscr L(ρ_t,\dotρ_t)=0$. We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when $\mathscr L$ is associated to the large deviations of a microscopic particle …
We prove that the quantum relative entropy is a rate function in large deviation principle. Next, we define information criteria for quantum states and estimate the accuracy of the use of them. Most of the results in this paper are essentially based on Hiai-Ohya-Tsukada theorem.
