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This paper considers the family of invariant measures of Markovian mean-field interacting particle systems on a countably infinite state space and studies its large deviation asymptotics. The Freidlin–Wentzell quasipotential is the usual candidate rate function for the sequence of invariant measures indexed by the number of particles. The paper provides two counterexamples where the quasipotential is not the rate function. The quasipotential arises from finite-horizon considerations. However, there are certain barriers that cannot be surmounted easily in any finite time horizon, but these barriers can be crossed in the stationary regime. Consequently, the quasipotential is infinite at some points where the rate function is finite. After highlighting this phenomenon, the paper studies some sufficient conditions on a class of interacting particle systems under which one can continue to assert that the Freidlin–Wentzell quasipotential is indeed the rate function.

For a Markov chain {X?} with general state space S and f:S?R ?, the large deviation principle for {n ?1 ? ??=1 f(X?)} is proved under a condition on the chain which is weaker than uniform recurrence but stronger than geometric recurrence and an integrability condition on f , for a broad class of initial distributions. This result is extended to the case when f takes values in a separable Banach space. Assuming only geometric ergodicity and under a non-degeneracy condition, a local large deviation result is proved for bounded f. A central analytical tool is the transform kernel, whose required properties, including new results, are established. The rate function in the large deviation results is expressed in terms of the convergence parameter of the transform kernel.

We analyse the spatial inhomogeneities ('spatial clustering') in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes ('caustics') in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents (FTLEs). Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor ('fractal catastrophes') make a distinct and universal contribution to the distribution of spatial FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media.

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.

The Thermodynamic Formalism provides a rigorous mathematical framework for studying quantitative and qualitative aspects of dynamical systems. At its core, there is a variational principle that corresponds, in its simplest form, to the Maximum Entropy principle. It is used as a statistical inference procedure to represent, by specific probability measures (Gibbs measures), the collective behaviour of complex systems. This framework has found applications in different domains of science. In particular, it has been fruitful and influential in neurosciences. In this article, we review how the Thermodynamic Formalism can be exploited in the field of theoretical neuroscience, as a conceptual and operational tool, in order to link the dynamics of interacting neurons and the statistics of action potentials from either experimental data or mathematical models. We comment on perspectives and open problems in theoretical neuroscience that could be addressed within this formalism.

We describe a method for computing transport coefficients from the direct evaluation of large deviation functions. This method is general, relying on only equilibrium fluctuations, and is statistically efficient, employing trajectory based importance sampling. Equilibrium fluctuations of molecular currents are characterized by their large deviation functions, which are scaled cumulant generating functions analogous to the free energies. A diffusion Monte Carlo algorithm is used to evaluate the large deviation functions, from which arbitrary transport coefficients are derivable. We find significant statistical improvement over traditional Green–Kubo based calculations. The systematic and statistical errors of this method are analyzed in the context of specific transport coefficient calculations, including the shear viscosity, interfacial friction coefficient, and thermal conductivity.

An important objective of empirical research on treatment response is to provide decision makers with information useful in choosing treatments. This paper studies minimax-regret treatment choice using the sample data generated by a classical randomized experiment. Consider a utilitarian social planner who must choose among the feasible statistical treatment rules, these being functions that map the sample data and observed covariates of population members into a treatment allocation. If the planner knew the population distribution of treatment response, the optimal treatment rule would maximize mean welfare conditional on all observed covariates. The appropriate use of covariate information is a more subtle matter when only sample data on treatment response are available. I consider the class of conditional empirical success rules; that is, rules assigning persons to treatments that yield the best experimental outcomes conditional on alternative subsets of the observed covariates. I derive a closed-form bound on the maximum regret of any such rule. Comparison of the bounds for rules that condition on smaller and larger subsets of the covariates yields sufficient sample sizes for productive use of covariate information. When the available sample size exceeds the sufficiency boundary, a planner can be certain that conditioning treatment choice on more covariates is preferable (in terms of minimax regret) to conditioning on fewer covariates.

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.

Importance sampling has become an important tool for the computation of extreme quantiles and tail-based risk measures. For estimation of such nonlinear functionals of the underlying distribution, the standard efficiency analysis is not necessarily applicable. In this paper we therefore study importance sampling algorithms by considering moderate deviations of the associated weighted empirical processes. Using a delta method for large deviations, combined with classical large deviation techniques, the moderate deviation principle is obtained for importance sampling estimators of two of the most common risk measures: value at risk and expected shortfall.

Let $\\xi,\\xi_1,\\xi_2,\\ldots$ be a sequence of independent identically distributed $d$-dimensional random vectors with zero mean and unit covariance matrix. Let $S_0:=0,$ $S_n:=\\sum_{i=1}n}\\xi_i$, $n\\ge 1,$ and let $s_n=s_n(t),$ $0\\le t\\le 1,$ be a random polygon with nodes at the points $(\\frac{k}{n},\\frac{S_k}{x})$, $0\\le k\\le n$, where $x=x(n)\\gg\\sqrt{n}$, $x=o(n)$ as $n\\to \\infty$. We establish a moderately large deviation principle for $s_n$ which asserts that, for a broad class of measurable sets $B$ of continuous functions, one has $\\log {\\bf P}(s_n\\in B)\\sim -\\frac{x arrow up }{n}\\inf_{f\\in B}I(f),$ where $I(f):=\\left\\{ \\begin{array}{ll} \\textstyle\\frac{1}{2}\\int_0 arrow right |f'(t)| arrow up \\,dt &\\mbox{if}\\ f(0)=0,\\ f\\ \\mbox{is absolutely}\\\ %[-4pt] &\\hspace{57pt}\\mbox{continuous,}\\\ \\infty&\\mbox{otherwise}. \\end{array}\\right.$ We establish the moderately large deviation principle for homogeneous processes with independent increments as well.