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Birth–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.

This paper deals with a random walk $S_n:=\\xi_1+\\cdots+\\xi_n$, $n=0,1,\\ldots,$ in the $d$-dimensional Euclidean space ${\\mathbb R}^d$, where $S_0=0$ and $\\xi_k$ are independent identically distributed random vectors satisfying Cramer's moment conditions. For random polygons with nodes at the points $(\\frac{k}{n},\\frac{1}{x}S_k)$, $k=0,1,\\ldots,n,$ we obtain the logarithmic asymptotics of the large deviation probabilities in different trajectory spaces when $x\\sim \\alpha_0 n$, $\\alpha_0>0$, as $n\\to\\infty.$ The results include the so-called local and extended large deviation principles (l.d.p.'s) (see ite15) that hold in those cases where the \"usual\" l.d.p. does not apply. The paper consists of three parts. Part I has two sections. Section 1 presents the key concepts and some facts concerning the l.d.p. in arbitrary metric spaces. In section 2 we formulate the \"strong\" versions of the \"usual\" l.d.p. in the large deviation zones that were obtained earlier in [A. A. Borovkov, Theory Probab. Appl., 12 (1967), pp. 575--595], [A. A. Mogul'skii, Theory Probab. Appl., 21 (1976), pp. 300--315] for the space of continuous functions. Besides that, section 2 also contains the l.d.p. for probabilities for the random walk trajectories to hit a convex set. That result was obtained using inequalities from [A. A. Borovkov and A. A. Mogul'skii, Theory Probab. Appl., 56 (2012), pp. 21--43] and does not involve any moment conditions. Part II begins with section 3 presenting an example elucidating the need to extend both the problem formulation and the very concept of the \"large deviation principle.\" We introduce a new extended functional space, a metric therein, and the deviation functional (integral) of a more general kind that will be used when constructing an \"extended\" l.d.p. In section 4 we present and prove the key results of the paper, the local and the extended large deviation principles, for the trajectories of univariate random walks in the space ${\\mathbb D}$ of functions without discontinuities of the second kind. Section 5 extends to the multivariate case all the results established in section 4. Section 6 in Part III presents results analogous to those from section 4, but now established in the space of functions of bounded variation with a metric stronger than that in $\\mathbb D$. In section 7 we establish the so-called conditional large deviation principles for the trajectories of univariate random walks given the location of the walk at the terminal point. As a consequence, we obtain the Sanov's theorem on large deviations of empirical distributions. [PUBLICATION ABSTRACT]

We complete the investigation of the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction which was started by Jahnel and one of the authors in (Sharp thresholds for Gibbsnon-Gibbs transitions in the fuzzy Potts model with a Kac-type interaction (2017)). As our main result of the present paper, we extend the previous sharpness result of mean-field bounds to cover all possible cases of fuzzy transformations, allowing also for the occurrence of Ising classes (containing precisely two spin values). The closing of this previously left open Ising-gap involves an analytical argument showing uniqueness of minimizing profiles for certain non-homogeneous conditional variational problems.

We investigate the Gibbs properties of the fuzzy Potts model on the d-dimensional torus with Kac interaction. We use a variational approach for profiles inspired by that of Fernández, den Hollander and Martínez [J. Stat. Phys. 156 (2014) 203–220] for their study of the Gibbs–non-Gibbs transitions of a dynamical Kac–Ising model on the torus. As our main result, we show that the mean-field thresholds dividing Gibbsian from non-Gibbsian behavior are sharp in the fuzzy Kac–Potts model with class size unequal two. On the way to this result, we prove a large deviation principle for color profiles with diluted total mass densities and use monotocity arguments.

We study a sequence of symmetric n-player stochastic differential games driven by both idiosyncratic and common sources of noise, in which players interact with each other through their empirical distribution. The unique Nash equilibrium empirical measure of the n-player game is known to converge, as n goes to infinity, to the unique equilibrium of an associated mean field game. Under suitable regularity conditions, in the absence of common noise, we complement this law of large numbers result with nonasymptotic concentration bounds for the Wasserstein distance between the n-player Nash equilibrium empirical measure and the mean field equilibrium. We also show that the sequence of Nash equilibrium empirical measures satisfies a weak large deviation principle, which can be strengthened to a full large deviation principle only in the absence of common noise. For both sets of results, we first use the master equation, an infinite-dimensional partial differential equation that characterizes the value function of the mean field game, to construct an associated McKean–Vlasov interacting n-particle system that is exponentially close to the Nash equilibrium dynamics of the n-player game for large n, by refining estimates obtained in our companion paper. Then we establish a weak large deviation principle for McKean–Vlasov systems in the presence of common noise. In the absence of common noise, we upgrade this to a full large deviation principle and obtain new concentration estimates for McKean–Vlasov systems. Finally, in two specific examples that do not satisfy the assumptions of our main theorems, we show how to adapt our methodology to establish large deviations and concentration results.

The present paper is a continuation of [Theory Probab. Appl., 57 (2013), pp. 1--27]. It consists of two sections. Section 6 presents results similar to those obtained in sections 4 and 5, but now in the space of functions of bounded variation with metric stronger than that of $\\D$. In section 7 we obtain the so-called conditional large deviation principles for the trajectories of univariate random walks with a localized terminal value of the walk. As a consequence, we prove a version of Sanov's theorem on large deviations of empirical distributions. [PUBLICATION ABSTRACT]

The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl. , 56 (2012), pp. 538--561]. It consists of three sections. Section 3 presents an example showing that it is necessary to extend the problem setup and the very concept of the \"large deviations principle\" (l.d.p.). We introduce a new, extended function space, a metric in it, and a deviation functional (integral) of a more general form (compared to the usual one) that will be used to construct an \"extended'' l.d.p. In section 4 we present and prove the main results of the paper for trajectories of univariate random walks in the space ${\\mathbb D}$ of functions without discontinuities of the second kind: the local and extended l.d.p.'s. Section 5 extends all the results from section 4 to the multivariate case.

We obtain analogues of the well-known Chebyshev's exponential inequality ${\\bf P}(\\xi \\ge x)\\le e^{-\\Lambda^{(\\xi)}(x)}$, $x>{\\bf E}\\,\\xi,$ for the distribution of a random variable $\\xi$, where $\\Lambda^{(\\xi)}(x):=\\sup_\\lambda\\{\\lambda x- \\log {\\bf E}\\,e^{\\lambda \\xi}\\}$ is the large deviation rate function for $\\xi$. Generalizations of this relation are established for multivariate random vectors $\\xi$, for sums of the vectors, and for trajectories of random processes associated with such sums.

We show two Freidlin–Wentzell-type Large Deviations Principles (LDP) in path space topologies (uniform and Hölder) for the solution process of McKean–Vlasov Stochastic Differential Equations (MV-SDEs) using techniques which directly address the presence of the law in the coefficients and altogether avoiding decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MV-SDEs having random coefficients and drifts of superlinear growth. As an application of our results, we establish a functional Strassen-type result (law of iterated logarithm) for the solution process of a MV-SDE.

We study Coulomb gases in any dimension d ≥ 2 and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes after extraction, generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity and screening, depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies.