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Renaud Raquépas

Publications and source records attributed to Renaud Raquépas.

At least 19 recordsLinked to original sources
arXiv2026

On the conservation of specific energy and entropy in infinite anharmonic systems

We work with infinite, closed, translation-invariant, finite-range lattice systems with "unbounded classical spins", also known as anharmonic crystals, under assumptions close to those used by Lanford, Lebowitz and Lieb (J. Stat. Phys., 1977); among other conditions, the pinning dominates the interaction. In this context, we prove conservation of the specific energy and specific entropy under the time evolution, and we discuss their relation to approach to thermal equilibrium, paralleling known results in the theory of quantum spin systems, where noncommutativity, as opposed to lack of compactness, is the main source of difficulties.

math-ph↗
arXiv2025

Computing large deviation rate functions of entropy production for diffusion processes by an interacting particle method

We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient to obtain the rate function is the computation of the principal eigenvalue $λ$ of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of $λ$ converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first one to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions.

math.NA↗
arXiv2025

Random attractors and nonergodic attractors for diffusions with degeneracies

We consider a diffusion on a bounded domain, assuming that the system is irreducible inside the domain and that the diffusion has varying degree of degeneracy on the domain's boundary. The long-term statistical properties of typical trajectories started inside the domain may be governed by one invariant measure or more than one invariant measure. We describe various possible scenarios. In dimensions 1 and 2 under boundary hyperbolicity assumptions, we give a complete classification of the limiting behavior and answer the question whether sequential averaging involving more than one invariant distribution occurs. In all cases, we compute the set of weak limit points of empirical measures. Our hitting-time estimates used to prove transience or recurrence are based on a new version of the Foster-Lyapunov technique. Extensions to nonhyperbolic boundaries and higher dimensions are discussed and an application to growth rates in scalable networks is given.

math.PR↗
arXiv2025

Divergence-free drifts decrease concentration

We show that bounded divergence-free vector fields $u : [0,\infty) \times \mathbb{R}^d \to\mathbb{R}^d$ decrease the ''concentration'', quantified by the modulus of absolute continuity with respect to the Lebesgue measure, of solutions to the associated advection-diffusion equation when compared to solutions to the heat equation. In particular, for symmetric decreasing initial data, the solution to the advection-diffusion equation has (without a prefactor constant) larger variance, larger entropy, and smaller $L^p$ norms for all $p \in [1,\infty]$ than the solution to the heat equation. We also note that the same is not true on $\mathbb{T}^d$.

math.AP↗
arXiv2024

Ziv-Merhav estimation for hidden-Markov processes

We present a proof of strong consistency of a Ziv-Merhav-type estimator of the cross entropy rate for pairs of hidden-Markov processes. Our proof strategy has two novel aspects: the focus on decoupling properties of the laws and the use of tools from the thermodynamic formalism.

cs.IT↗
arXiv2024

Large deviations of return times and related entropy estimators on shift spaces

We prove the large deviation principle for several entropy and cross entropy estimators based on return times and waiting times on shift spaces over finite alphabets. We consider shift-invariant probability measures satisfying some decoupling conditions which imply no form of mixing nor ergodicity. We establish precise relations between the rate functions of the different estimators, and between these rate functions and the corresponding pressures, one of which is the Rényi entropy function. For the most commonly used definition of return times, the large-deviation rate function is proved to be nonconvex, except in marginal cases. The results apply in particular to irreducible Markov chains, equilibrium measures for Bowen-regular potentials, g-measures, invariant Gibbs states for absolutely summable interactions in statistical mechanics, and also to probability measures which may be far from Gibbsian, including some hidden Markov models and repeated quantum measurement processes.

math.PR↗
arXiv2024

On the Ziv-Merhav theorem beyond Markovianity II: leveraging the thermodynamic formalism

We prove asymptotic results for a modification of the cross-entropy estimator originally introduced by Ziv and Merhav in the Markovian setting in 1993. Our results concern a more general class of decoupled measures on shift spaces over a finite alphabet and in particular imply strong asymptotic consistency of the modified estimator for all pairs of functions of stationary, irreducible, finite-state Markov chains satisfying a mild decay condition. Our approach is based on the study of a rescaled cumulant-generating function called the cross-entropic pressure, importing to information theory some techniques from the study of large deviations within the thermodynamic formalism.

math.PR↗
arXiv2023

On the Ziv-Merhav theorem beyond Markovianity

We generalize to a broader class of decoupled measures a result of Ziv and Merhav on universal estimation of the specific cross (or relative) entropy for a pair of multi-level Markov measures. The result covers pairs of suitably regular g-measures and pairs of equilibrium measures arising from the small space of interactions in mathematical statistical mechanics.

cs.IT↗
arXiv2022

Recurrence times, waiting times and universal entropy production estimators

The universal typical-signal estimators of entropy and cross entropy based on the asymptotics of recurrence and waiting times play an important role in information theory. Building on their construction, we introduce and study universal typical-signal estimators of entropy production in the context of nonequilibrium statistical mechanics of one-sided shifts over finite alphabets.

math-ph↗
arXiv2022

A gapped generalization of Kingman's subadditive ergodic theorem

We state and prove a generalization of Kingman's ergodic theorem on a measure-preserving dynamical system $(X,\mathcal{F},μ,T)$ where the $μ$-almost sure subadditivity condition $f_{n+m} \leq f_n + f_m \circ T^{n}$ is relaxed to a $μ$-almost sure, "gapped", almost subadditivity condition of the form $f_{n+σ_m+m} \leq f_n +ρ_n + f_m \circ T^{n+σ_n}$ for some nonnegative $ρ_n \in L^1(\mathrm{d}μ)$ and $σ_n \in \mathbf{N} \cup \{0\}$ that are suitably sublinear in $n$. This generalization has a first application to the existence of specific relative entropies for suitably decoupled measures on one-sided shifts.

math.DS↗
arXiv2022

The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

We investigate the behaviour of a family of entropy production functionals associated to stochastic differential equations of the form $\mathrm{d} X_s = -\nabla V(X_s) \, \mathrm{d} s + b(X_s) \, \mathrm{d} s + \sqrt{2ε} \, \mathrm{d} W_s $, where $b$ is a globally Lipschitz nonconservative vector field keeping the system out of equilibrium, with emphasis on the large-time limit and then the vanishing-noise limit. Different members of the family correspond to different choices of boundary terms. Our analysis yields a law of large numbers and a local large deviation principle which does not depend on the choice of boundary terms and which exhibits a Gallavotti--Cohen symmetry. We use techniques from the theory of semigroups and from semiclassical analysis to reduce the description of the asymptotic behaviour of the functional to the study of the leading eigenvalue of a quadratic approximation of a deformation of the infinitesimal generator near critical points of $V$.

math-ph↗
arXiv2022

On a waiting-time result of Kontoyiannis: mixing or decoupling?

We introduce conditions of lower decoupling to the study of waiting-time estimations of the cross entropy between two mutually independent stationary stochastic processes. Although similar decoupling conditions have been used in the literature on large deviations and statistical mechanics, they appear largely unexplored in information theory. Building on a result of Kontoyiannis, namely Theorem 4 in [Kontoyiannis, J. Theor. Probab., 1998], and replacing the $ψ$-mixing condition in this result with a lower decoupling condition, we considerably extend the validity of waiting-time estimation of cross entropy.

math.PR↗
arXiv2021

Fermionic walkers driven out of equilibrium

We consider a discrete-time non-Hamiltonian dynamics of a quantum system consisting of a finite sample locally coupled to several bi-infinite reservoirs of fermions with a translation symmetry. In this setup, we compute the asymptotic state, mean fluxes of fermions into the different reservoirs, as well as the mean entropy production rate of the dynamics. Formulas are explicitly expanded to leading order in the strength of the coupling to the reservoirs.

math-ph↗
arXiv2021

Exponential mixing under controllability conditions for SDEs driven by a degenerate Poisson noise

We prove existence and uniqueness of the invariant measure and exponential mixing in the total-variation norm for a class of stochastic differential equations driven by degenerate compound Poisson processes. In addition to mild assumptions on the distribution of the jumps for the driving process, the hypotheses for our main result are that the corresponding control system is dissipative, approximately controllable and solidly controllable. The solid controllability assumption is weaker than the well-known parabolic Hörmander condition and is only required from a single point to which the system is approximately controllable. Our analysis applies to Galerkin projections of stochastically forced parabolic partial differential equations with asymptotically polynomial nonlinearities and to networks of quasi-harmonic oscillators connected to different Poissonian baths.

math.PR↗
arXiv2019

On Fermionic walkers interacting with a correlated structured environment

We study the large-time behaviour of a sample $\mathcal{S}$ consisting of an ensemble of fermionic walkers on a graph interacting with a structured infinite reservoir of fermions $\mathcal{E}$ through an exchange of particles in preferred states. We describe the asymptotic state of $\mathcal{S}$ in terms the initial state of $\mathcal{E}$, with especially simple formulae in the limit of small coupling strength. We also study the particle fluxes into the different parts of the reservoir.

math-ph↗
arXiv2018

A note on Harris' ergodic theorem, controllability and perturbations of harmonic networks

We show that elements of control theory, together with an application of Harris' ergodic theorem, provide an alternate method for showing exponential convergence to a unique stationary measure for certain classes of networks of quasi-harmonic classical oscillators coupled to heat baths. With the system of oscillators expressed in the form $\mathrm{d} X_t = A X_t \,\mathrm{d} t + F(X_t) \,\mathrm{d} t + B \,\mathrm{d} W_t$ in $\mathbf{R}^d$, where $A$ encodes the harmonic part of the force and $-F$ corresponds to the gradient of the anharmonic part of the potential, the hypotheses under which we obtain exponential mixing are the following: $A$ is dissipative, the pair $(A,B)$ satisfies the Kalman condition, $F$ grows sufficiently slowly at infinity (depending on the dimension $d$), and the vector fields in the equation of motion satisfy the weak Hörmander condition in at least one point of the phase space.

math-ph↗
arXiv2018

Control of fluctuations and heavy tails for heat variation in the two-time measurement framework

We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0. On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation.

math-ph↗