We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction $X+Y \to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent simple continuous time random walks of total jump rate $2$. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval $[0,v]$, where $v$ is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state.
In this paper, we prove a central limit theorem and establish a moderate deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity. The proof for moderate deviation principle is based on the weak convergence approach.
In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short time. The proof for large deviation principle is based on the weak convergence approach. For small time asymptotics we use the exponential equivalence to prove the result.
In this paper, we establish a large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas in [5] plays a key role.
In this paper, we established a large deviation principle for stochastic models of incompressible second grade fluids. The weak convergence method introduced by \cite{Budhiraja-Dupuis} plays an important role.
Uniform large deviation principles for positive functionals of all equivalent types of infinite dimensional Brownian motions acting together with a Poisson random measure are established. The core of our approach is a variational representation formula which for an infinite sequence of i.i.d real Brownian motions and a Poisson random measure was shown in [5].
In this paper, we prove a central limit theorem and estabilish a moderate deviation principle for stochastic models of incompressible second fluids. The weak convergence method inreoduced by [4] plays an important role.
We prove an energy estimate for the polar empirical measure of the two-dimensional symmetric simple exclusion process. We deduce from this estimate and from results in reference [2] large deviations principles for the polar empirical measure and for the occupation time of the origin.
In this paper, we establish a moderate deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. We will adopt the weak convergence approach. Because of the appearance of jumps, this result is significantly different from that in Gaussian case.
The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles…
We are dealing with the validity of a large deviation principle for the two-dimensional Navier-Stokes equation, with periodic boundary conditions, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\e$ and $\d(\e)$, respectively, with $0<\e,\ \d(\e)<<1$. Depending on the relationship between $\e$ and $\d(\e)$ we will prove the validity of the large deviation principle in different functional spaces.
Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten (1984) obtained for boxes of particular orientation.
We derive properties of the rate function in Varadhan's (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
We study the large deviations of one-dimensional excited random walks. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n < nv)$ for $v \in (0,v_0)$ decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order $…
We establish the large deviation principle for solutions of one-dimensional SDEs with discontinuous coefficients. The main statement is formulated in a form similar to the classical Wentzel--Freidlin theorem, but under the considerably weaker assumption that the coefficients have no discontinuities of the second kind.
We demonstrate the large deviation principle in the small noise limit for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation. In this paper, we first prove the well-posedness of weak solutions to this system by the method of monotonicity. As we know, a recently developed method, weak convergent method, has been employed in studying the large deviations and this method is essentially based on the main result of \cite{ba2} which discloses the variational representation of exponential integrals with respect to the Brownian noise. The Itô inequality a…
We consider an exclusion process representing a reactive dynamics of a pulled front on the integer lattice, describing the dynamics of first class $X$ particles moving as a simple symmetric exclusion process, and static second class $Y$ particles. When an $X$ particle jumps to a site with a $Y$ particle, their position is intechanged and the $Y$ particle becomes an $X$ one. Initially, there is an arbitrary configuration of $X$ particles at sites $..., -1,0$, and $Y$ particles only at sites $1,2,...$, with a product Bernoulli law of parameter $ρ,0<ρ<1$. We prove a law of large numbers and a cen…
The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative …
The present work is concerned about two-dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations perturbed by a white noise (non degenerate) in smooth bounded domains in $\R^{2}$. We establish two important properties of the Markov semigroup associated with the solutions of 2D SCBF equations (for the absorption exponent $r=1,2,3$), that is, irreducibility and strong Feller property. These two properties implies the uniqueness of invariant measures and ergodicity also. Then, we discuss about the ergodic behavior of 2D SCBF equations by providing a Large Deviation Principle (LD…
We consider stochastic wave map equation on real line with solutions taking values in a $d$-dimensional compact Riemannian manifold. We show first that this equation has unique, global, strong in PDE sense, solution in local Sobolev spaces. The main result of the paper is a proof of the Large Deviations Principle for solutions in the case of vanishing noise.
Let $\mathcal{T}_n$ be the cover time of two-dimensional discrete torus $\mathbb{Z}^2_n=\mathbb{Z}^2/n\mathbb{Z}^2$. We prove that $\mathbb{P}[\mathcal{T}_n\leq \frac{4}πγn^2\ln^2 n]=\exp(-n^{2(1-\sqrtγ)+o(1)})$ for $γ\in (0,1)$. One of the main methods used in the proofs is the decoupling of the walker's trace into independent excursions by means of soft local times.
Random matrices have played an important role in many fields including machine learning, quantum information theory and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few of works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they…
We jointly investigate the existence of quasi-stationary distributions for one dimensional Lévy processes and the existence of traveling waves for the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation associated with the same motion. Using probabilistic ideas developed by S. Harris, we show that the existence of a traveling wave for the F-KPP equation associated with a centered Lévy processes that branches at rate $r$ and travels at velocity $c$ is equivalent to the existence of a quasi-stationary distribution for a Lévy process with the same movement but drifted by $-c$ and killed at zer…
We study the diffusion front for a natural two-dimensional model where many particles starting at the origin diffuse independently. It turns out that this model can be described using properties of near-critical percolation, and provides a natural example where critical fractal geometries spontaneously arise.
We lead the algorithm of expansion of sojourn probability of many-dimensional diffusion processes in small domain. The principal member of this expansion defines normalizing coefficient for special limit theorems.
We study the large deviations principle for one dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process $X_{t}$ in $\mathbb{R}$ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator $D_{v}D_{u}$, where $v$ and $u$ are two strictly increasing functions, $v$ is right continuous and $u$ is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is $εD_{v}D_{u}$ where $0<ε\ll…
We study the dynamics of smooth interval maps with non-flat critical points. For every such a map that is topologically exact, we establish the full (level-2) Large Deviation Principle for empirical means. In particular, the Large Deviation Principle holds for every non\nobreakdash-renormalizable quadratic map. This includes the maps without physical measure found by Hofbauer and Keller, and challenges the widely-shared view of the Large Deviation Principle as a refinement of laws of large numbers.
In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-$2$ large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincaré-Maxwell-Borel lemma, we deduce a conditional limit theorem for high-dimensional Orlicz balls. Roughly speaking, the latter shows that if $V_1$ and $V_2$ are Orlicz functions, then random points in the $V_1$-Orlicz ball, conditioned on having a small $V_2$-Orlicz radius, look like an appropriat…
We derive a large deviations principle for the two-dimensional two-component plasma in a box. As a consequence, we obtain a variational representation for the free energy, and also show that the macroscopic empirical measure of either positive or negative charges converges to the uniform measure. An appendix, written by Wei Wu, discusses applications to the supercritical complex Gaussian multiplicative chaos.
We establish large deviation principles for the couple of the maximum likelihood estimators of dimensional and drift coefficients in the generalised squared radial Ornstein-Uhlenbeck process. We focus our attention to the most tractable situation where the dimensional parameter $a>2$ and the drift parameter $b<0$. In contrast to the previous literature, we state large deviation principles when both dimensional and drift coefficient are estimated simultaneously.
We study the spherical model of a ferromagnet in $d$-dimensional cubes $Ω_n$ of volume $|Ω_n|=n^d$ and investigate large deviations of the magnetization of various domains $D_k\subset Ω_n$. We focus our attention on the low-temperature regime, $T<T_c$, and consider domains $D_k$ of three types: $(d-1)$-dimensional layers of width $k$, $(d-2)$-dimensional rods, and Kadanoff blocks. In the case of layers the large-deviation probabilities decay exponentially with $n^{d-2}$, and we obtain an explicit expression for the corresponding rate function. When the layer width $k\ll n$, the large-deviation…
The paper provides a description of the large deviation behavior for the Euclidean norm of projections of $\ell_p^n$-balls to high-dimensional random subspaces. More precisely, for each integer $n\geq 1$, let $k_n\in\{1,\ldots,n-1\}$, $E^{(n)}$ be a uniform random $k_n$-dimensional subspace of $\mathbb R^n$ and $X^{(n)}$ be a random point that is uniformly distributed in the $\ell_p^n$-ball of $\mathbb R^n$ for some $p\in[1,\infty]$. Then the Euclidean norms $\|P_{E^{(n)}}X^{(n)}\|_2$ of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension $n$ tend…
Given an $n$-dimensional random vector $X^{(n)}$ , for $k < n$, consider its $k$-dimensional projection $\mathbf{a}_{n,k}X^{(n)}$, where $\mathbf{a}_{n,k}$ is an $n \times k$-dimensional matrix belonging to the Stiefel manifold $\mathbb{V}_{n,k}$ of orthonormal $k$-frames in $\mathbb{R}^n$. For a class of sequences $\{X^{(n)}\}$ that includes the uniform distributions on scaled $\ell_p^n$ balls, $p \in (1,\infty]$, and product measures with sufficiently light tails, it is shown that the sequence of projected vectors $\{\mathbf{a}_{n,k}^\intercal X^{(n)}\}$ satisfies a large deviation principle…
