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Loïc Hervé

Publications and source records attributed to Loïc Hervé.

At least 19 recordsLinked to original sources
arXiv2023

Fine asymptotic expansion of the ODE's flow

In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the nonlinear ODE: X (t, x) = b X(t, x) with X(0, x) = x $\in$ R d , where b is a regular Z dperiodic vector field in R d. More precisely, we provide various conditions on b to obtain a "fine" asymptotic expansion of X of the type: |X(t, x) -- x -- t $ζ$(x)| $\le$ M < $\infty$, which is uniform with respect to t $\ge$ 0 and x $\in$ R d (or at least in a subset of R d), and where $ζ$(x) for x $\in$ R d , are the rotation vectors induced by the flow X. On the one hand, we give a necessary and sufficient condition on the vector field b so that the expansion X(t, x) -- x -- t $ζ$(x) reads as $Φ$ X(t, x) -- $Φ$(x), which yields immediately the desired expansion when the vector-valued function $Φ$ is bounded. In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on Y d and of vector-valued functions $Φ$. On the other hand, assuming that the two-dimensional Kolmogorov theorem and some extension in higher dimension hold, we establish different regimes depending on the commensurability of the rotation vectors of the flow X for which the fine estimate expansion of X is valid or not. It turns out that for any two-dimensional flow X associated with a non vanishing smooth vector field b and inducing a unique incommensurable rotation vector $ξ$, the fine asymptotic expansion of X holds in R 2 if, and only if, $ξ$ 1 /$ξ$ 2 is a Diophantine number. This result seems new in the setting of the ODE's flow. The case of commensurable rotation vectors $ζ$(x) is investigated in a similar way. Finally, several examples and counterexamples illustrate the different results of the paper, including the case of a vanishing vector field b which blows up the asymptotic expansion in some direction.

math.AP↗
arXiv2021

Specific properties of the ODE's flow in dimension two versus dimension three

This paper deals with the asymptotics of the ODE's flow induced by a regular vector field b on the d-dimensional torus R d /Z d. First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of R 2. Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field b = a $ζ$, where $ζ$ is a constant vector in R 2. Furthermore, several extensions of the

math.AP↗
arXiv2021

Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications

This paper deals with the long time asymptotics X(t, x)/t of the flow X solution to the autonomous vector-valued ODE: X (t, x) = b(X(t, x)) for t $\in$ R, with X(0, x) = x a point of the torus Y d := R d /Z d. We assume that the vector field b reads as the product $ρ$ $Φ$, where $ρ$ : Y d $\rightarrow$ [0, $\infty$) is a non negative regular function and $Φ$ : Y d $\rightarrow$ R d is a non vanishing regular vector field. In this work, the singleton condition means that the rotation set C b composed of the average values of b with respect to the invariant probability measures for the flow X is a singleton {$ζ$}, or equivalently, that lim t$\rightarrow$$\infty$ X(t, x)/t = $ζ$ for any x $\in$ Y d. This combined with Liouville's theorem regarded as a divergence-curl lemma, first allows us to obtain the asymptotics of the flow X when b is a current field. Then, we prove a general perturbation result assuming that $ρ$ is the uniform limit in Y d of a positive sequence ($ρ$ n) n$\in$N satisfying for any n $\in$ N, $ρ$ $\le$ $ρ$ n and C $ρ$n$Φ$ is a singleton {$ζ$ n }. It turns out that the limit set C b either remains a singleton, or enlarges to the closed line set [0, lim n $ζ$ n ] of R d. We provide various corollaries of this perturbation result involving or not the classical ergodic condition, according to the positivity or not of some harmonic means of $ρ$. These results are illustrated by different examples which show that the perturbation result is limited to the scalar perturbation of $ρ$, and which highlight the alternative satisfied by the rotation set C b. Finally, we prove that the singleton condition allows us to homogenize in any dimension the linear transport equation induced by the oscillating velocity b(x/$ε$) beyond any ergodic condition satisfied by the flow X.

math.AP↗
arXiv2021

Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $\mathbb{Z}^d$-periodic vector field from $\mathbb{R}^d$ in $\mathbb{R}^d$.We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: (i) the everywhere asymptotics of the flow $X$, (ii) the almost-everywhere asymptotics of the flow $X$, (iii) the global rectification of the vector field $b$ in $Y_d$, (iv) the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, (v) the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, (vi) the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, (vii) the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

math.AP↗
arXiv2019

Revisiting the asymptotics of the flow for some dynamical systems on the torus

In this paper we study the large time asymptotics of the flow of a dynamical system $X'=b(X)$ posed in the $d$-dimensional torus. Rather than using the classical unique ergodicity condition which is not fulfilled if $b$ vanishes at different points, we only assume that the set of the averages of $b$ with respect to the invariant probability measures for the flow is reduced to a singleton. We also rewrite the Liouville theorem which holds for any invariant probability measure $μ$, namely $μ\,b$ is divergence free, as a divergence-curl formula satisfied by any regular periodic function. The combination of these two tools turns out to be a new approach to get the asymptotics for some flows. This allows us to obtain the desired asymptotics in any dimension when $b = a\,ξ$ with $a$ a possibly vanishing periodic nonnegative function and $ξ$ a nonzero vector in $R^d$, or when $b = A\nabla v$ with $A$ a periodic nonnegative symmetric matrix-valued function and $v$ a periodic function.

math.DS↗
arXiv2017

Multiplicative ergodicity of laplace transforms for additive functional of markov chains with application to age-dependent branching process

We study the exponential growth of bifurcating processes with ancestral dependence. We suppose here that the lifetimes of the cells are dependent random variables, that the numbers of new cells are random and dependent. Lifetimes and new cells's numbers are also assumed to be dependent. We illustrate our results by examples, including some Markov models. Our approach is related to the behaviour of the Laplace transform of nonnegative additive functional of Markov chains and require weak moment assumption (no exponential moment is needed).

math.PR↗
arXiv2016

A computable bound of the essential spectral radius of finite range Metropolis--Hastings kernels

Let $π$ be a positive continuous target density on $\mathbb{R}$. Let $P$ be the Metropolis-Hastings operator on the Lebesgue space $\mathbb{L}^2(π)$ corresponding to a proposal Markov kernel $Q$ on $\mathbb{R}$. When using the quasi-compactness method to estimate the spectral gap of $P$, a mandatory first step is to obtain an accurate bound of the essential spectral radius $r\_{ess}(P)$ of $P$. In this paper a computable bound of $r\_{ess}(P)$ is obtained under the following assumption on the proposal kernel: $Q$ has a bounded continuous density $q(x,y)$ on $\mathbb{R}^2$ satisfying the following finite range assumption : $|u| \textgreater{} s \, \Rightarrow\, q(x,x+u) = 0$ (for some $s\textgreater{}0$). This result is illustrated with Random Walk Metropolis-Hastings kernels.

math.PR↗
arXiv2016

Additional material on bounds of $\ell^2$-spectral gap for discrete Markov chains with band transition matrices

We analyse the $\ell^2(π)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $π$. This analysis is heavily based on: first the study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(π)})$ of $P\_{|\ell^2(π)}$ derived from Hennion's quasi-compactness criteria; second the connection between the spectral gap property (SG$\_2$) of $P$ on $\ell^2(π)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$) is shown to hold under the condition \[α\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\, 1. \] Moreover $r\_{ess}(P\_{|\ell^2(π)}) \leq α\_0$. Simple conditions on asymptotic properties of $P$ and of its invariant probability distribution $π$ to ensure that $α\_0\textless{}1$ are given. In particular this allows us to obtain estimates of the $\ell^2(π)$-geometric convergence rate of random walks with bounded increments. The specific case of reversible $P$ is also addressed. Numerical bounds on the convergence rate can be provided via a truncation procedure. This is illustrated on the Metropolis-Hastings algorithm.

math.PR↗
arXiv2015

Computable bounds of ${\ell}^2$-spectral gap for discrete Markov chains with band transition matrices

We analyse the $\ell^2(π)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $π$. This analysis is heavily based on: first the study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(π)})$ of $P\_{|\ell^2(π)}$ derived from Hennion's quasi-compactness criteria; second the connection between the Spectral Gap property (SG$\_2$) of $P$ on $\ell^2(π)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$) is shown to hold under the condition $ α\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\, 1 $ Moreover $r\_{ess}(P\_{|\ell^2(π)}) \leq α\_0$. Effective bounds on the convergence rate can be provided from a truncation procedure.

math.PR↗
arXiv2015

Multiplicative ergodicity of Laplace transforms for additive functional of Markov chains

We study properties of the Laplace transforms of non-negative additive functionals of Markov chains. We are namely interested in a multiplicative ergodicity property used in [18] to study bifurcating processes with ancestral dependence. We develop a general approach based on the use of the operator perturbation method. We apply our general results to two examples of Markov chains, including a linear autoregressive model. In these two examples the operator-type assumptions reduce to some expected finite moment conditions on the functional (no exponential moment conditions are assumed in this work).

math.PR↗
arXiv2014

Approximating Markov chains and V-geometric ergodicity via weak perturbation theory

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. This paper provides explicit connections between the $V$-geometric ergodicity of $P$ and that of finite-rank nonnegative sub-Markov kernels $\Pc_k$ approximating $P$. A special attention is paid to obtain an efficient way to specify the convergence rate for $P$ from that of $\Pc_k$ and conversely. Furthermore, explicit bounds are obtained for the total variation distance between the $P$-invariant probability measure and the $\Pc_k$-invariant positive measure. The proofs are based on the Keller-Liverani perturbation theorem which requires an accurate control of the essential spectral radius of $P$ on usual weighted supremum spaces. Such computable bounds are derived in terms of standard drift conditions. Our spectral procedure to estimate both the convergence rate and the invariant probability measure of $P$ is applied to truncation of discrete Markov kernels on $\X:=\N$.

math.PR↗
arXiv2013

Spectral analysis of Markov kernels and application to the convergence rate of discrete random walks

Let $\{X_n\}_{n\in\N}$ be a Markov chain on a measurable space $\X$ with transition kernel $P$ and let $V:\X\r[1,+\infty)$. The Markov kernel $P$ is here considered as a linear bounded operator on the weighted-supremum space $\cB_V$ associated with $V$. Then the combination of quasi-compactness arguments with precise analysis of eigen-elements of $P$ allows us to estimate the geometric rate of convergence $ρ_V(P)$ of $\{X_n\}_{n\in\N}$ to its invariant probability measure in operator norm on $\cB_V$. A general procedure to compute $ρ_V(P)$ for discrete Markov random walks with identically distributed bounded increments is specified.

math.PR↗
arXiv2013

Additional material on local limit theorem for finite Additive Markov Processes [HL13]

This paper proposes additional material to the main statements of [HL13] which are are recalled in Section~2. In particular an application of [Theorem~2.2,HL13] to Renewal Markov Processes is provided in Section~4 and a detailed checking of the assumptions of [Theorem~2.2,HL13] for the joint distribution of local times of a finite jump process is reported in Section~5. A uniform version of [Theorem~2.2,HL13] with respect to a compact set of transition matrices is given in Section~6 (see [Remark 2.4,HL13]). The basic material on the semigroup of Fourier matrices and the spectral approach used in [HL13] is recalled in Section~3 in order to obtain a good understanding of the properties involved in this uniform version.

math.PR↗
arXiv2013

A local limit theorem for densities of the additive component of a finite Markov Additive Process

In this paper, we are concerned with centered Markov Additive Processes $\{(X_t,Y_t)\}_{t\in\T}$ where the driving Markov process $\{X_t\}_{t\in\T}$ has a finite state space. Under suitable conditions, we provide a local limit theorem for the density of the absolutely continuous part of the probability distribution of $t^{-1/2}Y_t$ given $X_0$. The rate of convergence and the moment condition are the expected ones with respect to the i.i.d case. An application to the joint distribution of local times of a finite jump process is sketched.

math.PR↗
arXiv2012

Regular perturbation of V -geometrically ergodic Markov chains

In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with i.i.d. noises (with a non-standard probability density function) is obtained.

math.PR↗
arXiv2012

Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\cB_V,\|\cdot\|_V)$ of all the measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.

math.PR↗
arXiv2012

A uniform Berry--Esseen theorem on $M$-estimators for geometrically ergodic Markov chains

Let $\{X_n\}_{n\ge0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_n(α):=n^{-1}\sum_{k=1}^nF(α,X_{k-1},X_k)$, $α\in\mathcal{A}\subset \mathbb {R}$. Consider an $M$ estimator $\hatα_n$, that is, a measurable function of the observations satisfying $M_n(\hatα_n)\leq \min_{α\in\mathcal{A}}M_n(α)+c_n$ with $\{c_n\}_{n\geq1}$ some sequence of real numbers going to zero. Under some standard regularity and moment assumptions, close to those of the i.i.d. case, the estimator $\hatα_n$ satisfies a Berry--Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.

math.ST↗
arXiv2012

On the recurrence set of planar Markov Random Walks

In this paper, we investigate the properties of recurrent planar Markov random walks. More precisely, we study the set of recurrent points with the use of local limit theorems. The Nagaev-Guivarc'h spectral method provides several examples for which these local limit theorems are satisfied as soon as the (standard or non-standard) central limit theorem holds.

math.PR↗