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Register a new userLarge-deviation principles of switching Markov processes via Hamilton-Jacobi equations
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We prove pathwise large-deviation principles of switching Markov processes by exploiting the connection to associated Hamilton-Jacobi equations, following Jin Fengs and Thomas Kurtzs method. In the limit that we consider, we show how the large-deviation problem in path-space reduces to a spectral problem of finding principal eigenvalues. The large-deviation rate functions are given in action-integral form. As an application, we demonstrate how macroscopic transport properties of stochastic models of molecular motors can be deduced from an associated principal-eigenvalue problem. The precise characterization of the macroscopic velocity in terms of principal eigenvalues implies that breaking of detailed balance is necessary for obtaining transport. In this way, we extend and unify existing results about molecular motors and place them in the framework of stochastic processes and large-deviation theory.
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The aim of this paper is to prove a Large Deviation Principle (LDP) for cumulative processes also known as coumpound renewal processes. These processes cumulate independent random variables occuring in time interval given by a renewal process. Our result extends the one obtained in Lefevere et al. (2011) in the sense that we impose no specific dependency between the cumulated random variables and the renewal process. The proof is inspired from Lefevere et al. (2011) but deals with additional difficulties due to the general framework that is considered here. In the companion paper Cattiaux-Costa-Colombani (2021) we apply this principle to Hawkes processes with inhibition. Under some assumptions Hawkes processes are indeed cumulative processes, but they do not enter the framework of Lefevere et al. (2011).
Let $(a_k)_{kinmathbb N}$ be a sequence of integers satisfying the Hadamard gap condition $a_{k+1}/a_k>q>1$ for all $kinmathbb N$, and let $$ S_n(omega) = sum_{k=1}^ncos(2pi a_k omega),qquad ninmathbb N,;omegain [0,1]. $$ The lacunary trigonometric sum $S_n$ is known to exhibit several properties typical for sums of independent random variables. In this paper we initiate the investigation of large deviation principles (LDPs) for $S_n$. Under the large gap condition $a_{k+1}/a_ktoinfty$, we prove that $(S_n/n)_{ninmathbb N}$ satisfies an LDP with speed $n$ and the same rate function $tilde{I}$ as for sums of independent random variables with the arcsine distribution, but show that the LDP may fail to hold when we only assume the Hadamard gap condition. However, we prove that in the special case $a_k=q^k$ for some $qin {2,3,ldots}$, $(S_n/n)_{ninmathbb N}$ satisfies an LDP with speed $n$ and a rate function $I_q$ different from $tilde{I}$. We also show that $I_q$ converges pointwise to $tilde I$ as $qtoinfty$ and construct a random perturbation $(a_k)_{kinmathbb N}$ of the sequence $(2^k)_{kinmathbb N}$ for which $a_{k+1}/a_kto 2$ as $ktoinfty$, but for which $(S_n/n)_{ninmathbb N}$ satisfies an LDP with the rate function $tilde{I}$ as in the independent case and not, as one might na{i}vely expect, with rate function $I_2$. We relate this fact to the number of solutions of certain Diophantine equations. Our results show that LDPs for lacunary trigonometric sums are sensitive to the arithmetic properties of $(a_k)_{kinmathbb N}$. This is particularly noteworthy since no such arithmetic effects are visible in the central limit theorem by Salem and Zygmund or in the law of the iterated logarithm by Erdos and Gal. Our proofs use a combination of tools from probability theory, harmonic analysis, and dynamical systems.
Information-theory based variational principles have proven effective at providing scalable uncertainty quantification (i.e. robustness) bounds for quantities of interest in the presence of nonparametric model-form uncertainty. In this work, we combine such variational formulas with functional inequalities (Poincar{e}, $log$-Sobolev, Liapunov functions) to derive explicit uncertainty quantification bounds for time-averaged observables, comparing a Markov process to a second (not necessarily Markov) process. These bounds are well-behaved in the infinite-time limit and apply to steady-states of both discrete and continuous-time Markov processes.
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