Pierre Del Moral
email: pierre.del-moral@inria.fr

Peng Hu
email: peng.hu@inria.fr

Liming Wu
email: wuliming@amt.ac.cn

Keywords: Concentration properties, Feynman-Kac particle processes, stochastic particle models

These lecture notes present some new concentration inequalities for Feynman-Kac particle processes. We analyze different types of stochastic particle models, including particle profile occupation measures, genealogical tree based evolution models, particle free energies, as well as backward Markov chain particle models. We illustrate these results with a series of topics related to computational physics and biology, stochastic optimization, signal processing and bayesian statistics, and many other probabilistic machine learning algorithms. Special emphasis is given to the stochastic modeling and the quantitative performance analysis of a series of advanced Monte Carlo methods, including particle filters, genetic type island models, Markov bridge models, interacting particle Markov chain Monte Carlo methodologies.

Sur les propriétés concentrales de processus particulaires d'interaction

Résumé : Ces notes de cours présentent de nouvelles inégalités de concentration exponentielles pour les processus empiriques en interaction associés à des modèles particulaires de type Feynman-Kac. Nous analysons diffèrents modèles stochastiques, notamment des mesures d'occupation courante de population génétiques, des modèles historiques basés sur des évolutions d'arbres généalogiques, des estimations d'énergies libres, ainsi que des modèles de chaînes de Markov particulaires à rebours. Nous illustrons ces résultats avec une série d'applications liées à la physique numérique et la biologie, l'optimisation stochastique, le traitement du signal et la statistique bayésienne, avec de nombreux algorithmes probabilistes d'apprentissage automatique. Un accent particulier est donné à la modélisation stochastique de ces algorithmes de Monte Carlo, et à l'analyse quantitative de leurs performances. Nous examinons notamment la convergence de filtres particulaires, des "Island models" de type génétique, des processus de ponts markoviens, ainsi que diverses méthodes de type MCMC en interaction.

Mots-clés : Propriétés de concentration exponentielle, processus empiriques en interaction, interprétations particulaires de formules de Feynman-Kac.

1 Stochastic particle methods

Introduction

Stochastic particle methods have come to play a significant role in applied probability, numerical physics, Bayesian statistics, probabilistic machine learning, and engineering sciences. They are increasingly used to solve a variety of problems. To name a few, nonlinear filtering equations, data assimilation problems, rare event sampling, hidden Markov chain parameter estimation, stochastic control problems, financial mathematics. There are also used in computational physics for free energy computations, and Schrödinger operator's ground states estimation problems, as well as in computational chemistry for sampling the conformation of polymers in a given solvent.

Understanding rigorously these new particle Monte Carlo methodologies leads to fascinating mathematics related to Feynman-Kac path integral theory and their interacting particle interpretations [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF]. In the last two decades, this line of research has been developed by using methods from stochastic analysis of interacting particle systems and nonlinear semigroup models in distribution spaces, but it has also generated difficult questions that cannot be addressed without developing new mathematical tools.

Let us survey some of the important challenges that arise.

For numerical applications, it is essential to obtain non asymptotic quantitative information on the convergence of the algorithms. Asymptotic theory, including central limit theorems, moderate deviations, and large deviations principles have clearly limited practical values. An overview of these asymptotic results in the context of mean field and Feynman-Kac particle models can be found in the series of articles [START_REF] Dawson | Large deviations for interacting processes in the strong topology[END_REF][START_REF] Del Moral | Large deviations for interacting particle systems. Applications to nonlinear filtering problems[END_REF][START_REF] Del Moral | A central limit theorem for nonlinear filtering using interacting particle systems[END_REF][START_REF] Del Moral | The Monte-Carlo method for filtering with discrete-time observations: Central limit theorems[END_REF][START_REF] Moral | A note on Laplace-Varadhan's integral lemma[END_REF][START_REF] Douc | Moderate Deviations for Particle Filtering[END_REF].

Furthermore, when solving a given concrete problem, it is important to obtain explicit non asymptotic error bounds estimates to ensure that the stochastic algorithm is provably correct. While non asymptotic propagation of chaos results provide some insights on the bias properties of these models, they rarely provide useful effective convergence rates.

Last but not least, it is essential to analyze the robustness properties, and more particularly the uniform performance of particle algorithms w.r.t. the time horizon. By construction, these important questions are intimately related to the stability properties of complex nonlinear Markov chain semigroups associated with the limiting measure valued process. This line of ideas has been further developed in the articles [START_REF] Dawson | Large deviations for interacting processes in the strong topology[END_REF][START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF][START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Moral | Concentration inequalities for mean field particle models[END_REF], and in the books [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF].

Without any doubt, one of the most powerful mathematical tools to analyze the deviations of Monte Carlo based approximations is the theory of empirical processes and measure concentration theory. In the last two decades, these new tools have become one of the most important step forward in infinite dimensional stochastic analysis, advanced machine learning techniques, as well as in the development of a statistical non asymptotic theory.

In recent years, a lot of effort has been devoted to describing the behavior of the supremum norm of empirical functionals around the mean value of the norm. For an overview of these subjects, we refer the reader to the seminal books of D. Pollard [START_REF] Pollard | Empirical processes : theory and applications[END_REF], and the one of A.N. Van der Vaart and J.A. Wellner [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes with Applications to Statistics[END_REF], and the remarkable articles by E. Giné [START_REF] Giné | Lectures on some aspects of the bootstrap[END_REF], M. Ledoux [START_REF] Ledoux | The concentration of measure phenomenon[END_REF], and M. Talagrand [START_REF] Talagrand | Concentration of measure and isoperimetric inequalities in product spaces[END_REF][START_REF] Talagrand | A new look at independence[END_REF][START_REF] Talagrand | New concentration inequalities in product spaces[END_REF], and the more recent article by R. Adamczak [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF]. The best constants in Talagrand's concentration inequalities were obtained by Th. Klein, and E. Rio [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF]. In this article, the authors proved the functional version of Bennett's and Bernstein's inequalities for sums of independent random variables.

To main difficulties we encountered in applying these concentration inequalities to interacting particle models are of different order:

Firstly, all of the concentration inequalities developed in the literature on empirical processes still involve the mean value of the supremum norm empirical functionals. In practical situation, these tail style inequalities can only be used if we have some precise information on the magnitude of the mean value of the supremum norm of the functionals.

On the other hand, the range of application the theory of empirical processes and measure concentration theory is restricted to independent random samples, or equivalently product measures, and more recently to mixing Markov chain models. In the reverse angle, stochastic particle techniques are not based on fully independent sequences, nor on Markov chain Monte Carlo principles, but on interacting particle samples combined with complex nonlinear Markov chain semigroups. More precisely, besides the fact that particle models are built sequentially using conditionally independent random samples, their respective conditional distributions are still random. In addition, they strongly depend in a nonlinear way on the occupation measure of the current population.

In summary, the concentration analysis of interacting particle processes require the development of new stochastic perturbation style techniques to control the interaction propagation and the independence degree between the samples.

The first article extending empirical processes theory to particle models is a joint work of the first author with M. Ledoux [START_REF] Del Moral | On the Convergence and the Applications of Empirical Processes for Interacting Particle Systems and Nonlinear Filtering[END_REF]. In this work, we proved Glivenko-Cantelli and Donsker theorems under entropy conditions, as well as non asymptotic exponential bounds for Vapnik-Cervonenkis classes of sets or functions. Nevertheless, in practical situations these non asymptotic results tend to be a little disappointing, with very poor constants that degenerate w.r.t. the time horizon.

The second most important result on the concentration properties of mean field particle model is the recent article of the first author with E. Rio [START_REF] Moral | Concentration inequalities for mean field particle models[END_REF]. This article is only concerned with finite marginal model. The authors generalize the classical Hoeffding, Bernstein and Bennett inequalities for independent random sequences to interacting particle systems.

In these notes, we survey some of these results, and we provide new concentration inequalities for interacting empirical processes. We emphasize that these lectures don't give a comprehension treatment of the theory of interacting empirical processes. To name a few missing topics, we do not discuss large deviation principles w.r.t. the strong τ -topology, Donsker type fluctuation theorems, moderate deviation principles, and continuous time models. The first two topics and developed in the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], the third one is developed in [START_REF] Del Moral | Moderate deviations for mean field particle models[END_REF], the last one is still an open research subject.

These notes emphasize a single stochastic perturbation method, with second order expansion entering the stability properties of the limiting Feynman-Kac semigroups. The concentration results attained are probably not the best possible of their kind. We have chosen to strive for just enough generality to derive useful and uniform concentration inequalities w.r.t. the time horizon, without having to impose complex and often unnatural regularity conditions to squeeze them into the general theory of empirical processes. Some of the results are borrowed from the recent article [START_REF] Moral | Concentration inequalities for mean field particle models[END_REF], and many others are new. These notes should be complemented with the book [START_REF] Doucet | Sequential Monte Carlo Methods in Pratice[END_REF], the books [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF], and the article [START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF]. A very basic knowledge in statistics and machine learning theory will be useful, but not necessary. Good backgrounds in Markov chain theory and in stochastic semigroup analysis are necessary.

We have made our best to give a self-contained presentation, with detailed proofs assuming however some familiarity with Feynman-Kac models, and basic facts on the theory of Markov chains on abstract state spaces. Only in section 4.6.1, we have skipped the proof of some tools from convex analysis. We hope that the essential ideas are still accessible to the readers.

It is clearly not the scope of these lecture notes to give an exhaustive list of references to articles in computational physics, engineering sciences, and machine learning presenting heuristic like particle algorithms to solve a specific estimation problem. Up to a few exceptions, we have only provided references to articles with rigorous and well founded mathematical treatments on particle models. We already apologize for possible errors, or for references that have been omitted due to the lack of accurate information.

These notes grew from series of lectures the first author gave in the Computer Science and Communications Research Unit, of the University of Luxembourg in February and March 2011. They were reworked, with the addition of new material on the concentration of empirical processes for a course given at the Sino-French Summer Institute in Stochastic Modeling and Applications (CNRS-NSFC Joint institute of Mathematics), held at the Academy of Mathematics and System Science, Beijing, on June 2011. The Summer Institute was ably organized by Fuzhou Gong, Ying Jiao, Gilles Pagès, and Mingyu Xu, and the members of the scientific committee, including Nicole El Karoui, Zhiming Ma, Shige Peng, Liming Wu, Jia-An Yan, and Nizar Touzi. The first author is grateful to them for giving to him the opportunity to experiment on a receptive audience with material not entirely polished.

In reworking the lectures, we have tried to resist the urge to push the analysis to general classes of mean field particle models, in the spirit of the recent joint article with E. Rio [START_REF] Moral | Concentration inequalities for mean field particle models[END_REF]. Our principal objective has been to develop just enough analysis to handle four types of Feynman-Kac interacting particle processes; namely, genetic dynamic population models, genealogical tree based algorithms, particle free energies, as well as backward Markov chain particle models. These application models do not exhaust the possible uses of the theory developed in these lectures.

A brief review on particle algorithms

Stochastic particle methods belong to the class of Monte Carlo methods. They can be thought as an universal particle methodology for sampling complex distributions in highly dimensional state spaces.

We can distinguish two different classes of models; namely, diffusion type interacting processes, and interacting jump particle models. Feynman-Kac particle methods belongs to the second class of models, with rejection-recycling jump type interaction mechanisms. In contrast to conventional acceptancerejection type techniques, Feynman-Kac particle methods are equipped with an adaptive and interacting recycling strategy.

The common central feature of all the Monte Carlo particle methodologies developed so far is to solve discrete generation, or continuous time integrodifferential equations in distribution spaces. The first heuristic like description of these probabilistic techniques in mathematical physics goes back to the Los Alamos report [START_REF] Everett | Los Alamos report[END_REF], and the article [START_REF] Everett | [END_REF] by C.J. Everett and S. Ulam in 1948, and the short article by N. Metropolis and S. Ulam [START_REF] Metropolis | The Monte Carlo method[END_REF], published in 1949.

In some instances, the flow of measures is dictated by the problem at hand. In advanced signal processing, the conditional distributions of the signal given partial and noisy observations are given by the so-called nonlinear filtering equation in distribution space (see for instance [START_REF] Moral | Non Linear Filtering: Interacting Particle Solution[END_REF][START_REF] Moral | Measure Valued Processes and Interacting Particle Systems. Application to Non Linear Filtering Problems[END_REF][START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF], and references therein).

Free energies and Schrödinger operator's ground states are given by the quasi-invariant distribution of a Feynman-Kac conditional distribution flow of non absorbed particles in absorbing media. We refer the reader to the articles by E. Cancès, B. Jourdain and T. Lelièvre [START_REF] Cancès | Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation[END_REF], M. El Makrini, B. Jourdain and T. Lelièvre [START_REF] Makrini | Diffusion Monte Carlo method: Numerical analysis in a simple case[END_REF], M. Rousset [START_REF] Rousset | On the control of an interacting particle approximation of Schroedinger ground states[END_REF], the pair of articles of the first author with L. Miclo [START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF][START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF], the one with A. Doucet [START_REF] Del Moral | Particle Motions in Absorbing Medium with Hard and Soft Obstacles[END_REF], and the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], and the references therein.

In mathematical biology, branching processes and infinite population models are also expressed by nonlinear parabolic type integro-differential equations. Further details on this subject can be found in the articles by D.A. Dawson and his co-authors [START_REF] Dawson | Stochastic evolution equations and related measure processes[END_REF][START_REF] Dawson | Measure-valued Markov processes[END_REF][START_REF] Dawson | On weak convergence of branching particle systems undergoing spatial motion[END_REF], the works of E.B. Dynkin [START_REF] Dynkin | An introduction to branching measure-valued processes[END_REF], and J.F. Le Gall [START_REF] Gall | Spatial branching processes, random snakes, and partial different equations[END_REF], and more particularly the seminal book of S.N. Ethier and T.G. Kurtz [START_REF] Ethier | Markov processes: Characterization and convergence[END_REF], and the pioneering article by W. Feller [START_REF] Feller | Diffusion processes in genetics[END_REF].

In other instances, we formulate a given estimation problem in terms a sequence of distributions with increasing complexity on state space models with increasing dimension. These stochastic evolutions can be related to decreasing temperature schedules in Boltzmann-Gibbs measures, multilevel decompositions for rare event excursion models on critical level sets, decreasing subsets strategies for sampling tail style distributions, and many other sequential importance sampling plan. For a more thorough discussion on these models we refer the reader to [START_REF] Del Moral | Sequential Monte Carlo samplers[END_REF].

From the pure probabilistic point of view, any flow of probability measures can be interpreted as the evolution of the laws of the random states of a Markov process. In contrast to conventional Markov chain models, the Markov transitions of these chains may depend on the distribution of the current random state. The mathematical foundations of these discrete generation models have been started in 1996 in [START_REF] Moral | Non Linear Filtering: Interacting Particle Solution[END_REF] in the context of nonlinear filtering problems. Further analysis was developed in a joint work [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF] of the first author with L. Miclo published in 2000. For a more thorough discussion on the origin and the performance analysis of these discrete generation models, we also refer to the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], and the joint articles of the first author with A. Guionnet [START_REF] Del Moral | Large deviations for interacting particle systems. Applications to nonlinear filtering problems[END_REF][START_REF] Del Moral | A central limit theorem for nonlinear filtering using interacting particle systems[END_REF][START_REF] Del Moral | On the stability of Measure Valued Processes with Applications to filtering[END_REF][START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF], and M. Kouritzin [START_REF] Del Moral | On a class of discrete generation interacting particle systems[END_REF].

The continuous time version of these nonlinear type Markov chain models take their origins from the 1960s, with the development of fluid mechanisms and statistical physics. We refer the reader to the pioneering works of H.P. McKean [START_REF] Mckean | A class of Markov processes associated with nonlinear parabolic equations[END_REF][START_REF] Mckean | Propagation of chaos for a class of non-linear parabolic equations[END_REF], see also the more recent treatments by N. Bellomo and M. Pulvirenti [START_REF] Bellomo | Generalized kinetic models in applied sciences[END_REF][START_REF] Bellomo | Modeling in Applied Sciences[END_REF], the series of articles by C. Graham and S. Méléard on interacting jump models [START_REF] Graham | Stochastic particle approximations for generalized Boltzmann models and convergence estimates[END_REF][START_REF] Graham | Probabilistic tools and Monte-Carlo approximations for some Boltzmann equations[END_REF][START_REF] Pulvirenti | Kinetic limits for stochastic particle systems[END_REF], the articles by S. Méléard on Boltzmann equations [START_REF] Méléard | Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models[END_REF][START_REF] Méléard | Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations[END_REF][START_REF] Méléard | Probabilistic interpretation and approximations of some Boltzmann equations[END_REF][START_REF] Méléard | Stochastic approximations of the solution of a full Boltzmann equation with small initial data[END_REF], and the lecture notes of A.S. Sznitman [START_REF] Sznitman | Topics in propagation of chaos[END_REF], and references therein.

In contrast to conventional Markov chain Monte Carlo techniques, these McKean type nonlinear Markov chain models can be thought as perfect importance sampling strategies, in the sense that the desired target measures coincide at any time step with the law of the random states of a Markov chain. Unfortunately, as we mentioned above, the transitions of these chains depend on the distributions of its random states. Thus, they cannot be sampled without an additional level of approximation. One natural solution is to use a mean field particle interpretation model. These stochastic techniques belong to the class of stochastic population models, with free evolutions mechanisms, coupled with branching and/or adaptive interacting jumps. At any time step, the occupation measure of the population of individuals approximate the solution of the nonlinear equation, when the size of the system tends to ∞.

In genetic algorithms and sequential Monte Carlo literature, the reference free evolution model is interpreted as a reference sequence of twisted Markov chain samplers. These chains are used to perform the mutation/proposal transitions. As in conventional Markov chain Monte Carlo methods, the interacting jumps are interpreted as an acceptance-rejection transition, equipped with sophisticated interacting and adaptive recycling mechanism. In Bayesian statistics and engineering sciences, the resulting adaptive particle sampling model is often coined as a sequential Monte Carlo algorithm, genetic procedures, or simply Sampling Importance Resampling methods, mainly because it is based on importance sampling plans and online approximations of a flow of probability measures.

Since the 1960s, the adaptive particle recycling strategy has also been associated in biology and engineering science with several heuristic-like paradigms, with a proliferation of botanical names, depending the application area they are thought: bootstrapping, switching, replenishing, pruning, enrichment, cloning, reconfigurations, resampling, rejuvenation, acceptance/rejection, spawning.

Of course, the idea of duplicating online better-fitted individuals and moving them one step forward to explore state-space regions is the basis of various stochastic search algorithms. To name a few:

Particle and bootstrap filters, Rao-Blackwell particle filters, sequential Monte Carlo methods, sequentially Interacting Markov chain Monte Carlo, genetic type search algorithms, Gibbs cloning search techniques, interacting simulated annealing algorithms, sampling-importance resampling methods, quantum Monte Carlo walkers, adaptive population Monte Carlo sampling models, and many others evolutionary type Monte Carlo methods.

For a more detailed discussion on these models, with precise references we refer the reader to the three books [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF][START_REF] Doucet | Sequential Monte Carlo Methods in Pratice[END_REF].

Feynman-Kac path integrals

Feynman-Kac measures represent the distribution of the paths of a Markov process, weighted by a collection of potential functions. These functional models are natural mathematical extensions of the traditional changes of probability mea-sures, commonly used in importance sampling technologies, Bayesian inference, and in nonlinear filtering modeling.

These stochastic models are defined in terms of only two ingredients: A Markov chain X n , with Markov transition M n on some measurable state spaces (E n , E n ) with initial distribution η 0 , and a sequence of (0, 1]-valued potential functions G n on the set E n .

The Feynman-Kac path measure associated with the pairs (M n , G n ) is the probability measure Q n on the product state space

E n := (E 0 × . . . × E n )
defined by the following formula

dQ n := 1 Z n    0≤p<n G p (X p )    dP n (1)
where Z n is a normalizing constant and P n is the distribution of the random paths

X n = (X 0 , . . . , X n ) ∈ E n
of the Markov process X p from the origin p = 0 with initial distribution η 0 , up to the current time p = n. We also denote by

Γ n = Z n Q n (2)
its unnormalized version.

The prototype model we have in head is the traditional particle absorbed Markov chain model

X c n ∈ E c n := E n ∪ {c} absorption ∼(1-Gn) ------------------→ X c n exploration ∼Mn+1 ------------------→ X c n+1 (3)
The chain X c n starts at some initial state X c 0 randomly chosen with distribution η 0 . During the absorption stage, we set X c n = X c n with probability G n (X n ), otherwise we put the particle in an auxiliary cemetery state X c n = c. When the particle X c n is still alive (that is, if we have X c n ∈ E n ), it performs an elementary move X c n X c n+1 according to the Markov transition M n+1 . Otherwise, the particle is absorbed and we set X c p = X c p = c, for any time p > n. If we let T be the first time X c n = c, then we have the Feynman-Kac representation formulae

Q n = Law((X c 0 , . . . , X c n ) | T ≥ n) and Z n = Proba (T ≥ n)
For a more thorough discussion on the variety of application domains of Feynman-Kac models, we refer the reader to chapter 2.

We also denote by η n and γ n , the n-th time marginal of Q n and Γ n . It is a simple exercise to check that

γ n = γ n-1 Q n and η n+1 = Φ n+1 (η n ) := Ψ Gn (η n )M n+1 (4)
with the positive integral operator

Q n (x, dy) = G n-1 (x) M n (x, dy) RR n°7677
and the Boltzmann-Gibbs transformation

Ψ Gn (η n )(dx) = 1 η n (G n ) G n (x) η n (dx) (5)
In addition, the normalizing constants Z n can be expressed in terms of the flow of marginal measures η p , from the origin p = 0 up to the current time n, with the following multiplicative formulae:

Z n := γ n (1) = E   0≤p<n G p (X p )   = 0≤p<n η p (G p ) (6)
This multiplicative formula is easily checked using the induction

γ n+1 (1) = γ n (G n ) = η n (G n ) γ n (1)
The abstract formulae discussed above are more general than it may appear. For instance, they can be used to analyze without further work path spaces models, including historical processes or transition space models, as well as finite excursion models. These functional models also encapsulated quenched Feynman-Kac models, Brownian type bridges and linear Gaussian Markov chains conditioned on starting and end points.

For a more thorough discussion on these path space models, we refer the reader to section 2.4, section 2.6, chapters 11-12 in the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], as well as to the section 2, in the former lecture notes.

When the Markov transitions M n are absolutely continuous with respect to some measures λ n on E n , and for any (x, y) ∈ (E n-1 × E n ) we have

H n (x, y) := dM n (x, .) dλ n (y) > 0 (7)
we also have the following backward formula

Q n (d(x 0 , . . . , x n )) = η n (dx n ) n q=1 M q,ηq-1 (x q , dx q-1 ) (8)
with the the collection of Markov transitions defined by

M n+1,ηn (x, dy) ∝ G n (y) H n+1 (y, x) η n (dy) (9)
The proof of this formula is housed in section 3.2. Before launching into the description of the particle approximation of these models, we end this section with some connexions between discrete generation Feynman-Kac models and more conventional continuous time models arising in physics and scientific computing.

The Feynman-Kac models presented above play a central role in the numerical analysis of certain partial differential equations, offering a natural way to solve these functional integral models by simulating random paths of stochastic processes. These Feynman-Kac models were originally presented by Mark Kac in 1949 [START_REF] Kac | On distributions of certain Wiener functionals[END_REF] for continuous time processes.

These continuous time models are used in molecular chemistry and computational physics to calculate the ground state energy of some Hamiltonian

RR n°7677

operators associated with some potential function V describing the energy of a molecular configuration (see for instance [START_REF] Cancès | Quantum Monte Carlo simulations of fermions. A mathematical analysis of the fixed-node approximation[END_REF][START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Makrini | Diffusion Monte Carlo method: Numerical analysis in a simple case[END_REF][START_REF] Rousset | On the control of an interacting particle approximation of Schroedinger ground states[END_REF], and references therein). To better connect these partial differential equation models with [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF], let us assume that M n (x n-1 , dx n ) is the Markov probability transition X n = x n X n+1 = x n+1 coming from a discretization in time X n = X ′ tn of a continuous time E-valued Markov process X ′ t on a given time mesh (t n ) n≥0 with a given time step (t nt n-1 ) = ∆t. For potential functions of the form G n = e -V ∆t , the measures Q n ≃ ∆t→0 Q tn represents the time discretization of the following distribution:

dQ t = 1 Z t exp - t 0 V (X ′ s ) ds dP X ′ t
where P X ′ t stands for the distribution of the random paths (X ′ s ) 0≤s≤t with a given infinitesimal generator L. The marginal distributions γ t at time t of the unnormalized measures Z t dQ t are the solution of the so-called imaginary time Schroedinger equation, given in weak formulation on sufficiently regular function f by the following intregro-differential equation

d dt γ t (f ) := γ t (L V (f )) with L V = L -V
The errors introduced by the discretization of the time are well understood for regular models, we refer the interested reader to [START_REF] Del Moral | The Monte-Carlo Method for filtering with discrete-time observations[END_REF][START_REF] Di Masi | An approximation for the nonlinear filtering problem with error bounds[END_REF][START_REF] Korezlioglu | Filtering for nonlinear systems driven by nonwhite noises : An approximating scheme[END_REF][START_REF] Picard | Approximation of the non linear filtering problems and order of convergence[END_REF] in the context of nonlinear filtering.

Interacting particle systems

The stochastic particle interpretation of the Feynman-Kac measures (1) starts with a population of N candidate possible solutions (ξ 1 0 , . . . , ξ N 0 ) randomly chosen w.r.t. some distribution η 0 .

The coordinates ξ i 0 also called individuals or phenotypes, with 1 ≤ N . The random evolution of the particles is decomposed into two main steps : the free exploration and the adaptive selection transition.

During the updating-selection stage, multiple individuals in the current population (ξ 1 n , . . . , ξ N n ) at time n ∈ N are stochastically selected based on the fitness function G n . In practice, we choose a random proportion B i n of an existing solution ξ i n in the current population with a mean value ∝ G n (ξ i n ) to breed a brand new generation of "improved" solutions ( ξ 1 n , . . . , ξ N n ). For instance, for every index i, with a probability ǫ n G n (ξ i n ), we set ξ i n = ξ i n , otherwise we replace ξ i n with a new individual ξ i n = ξ j n randomly chosen from the whole population with a probability proportional to G n (ξ j n ). The parameter ǫ n ≥ 0 is a tuning parameter that must satisfy the constraint ǫ n G n (ξ i n ) ≤ 1, for every 1 ≤ i ≤ N . During the prediction-mutation stage, every selected individual ξ i n moves to a new solution ξ i n+1 = x randomly chosen in E n+1 , with a distribution M n+1 ( ξ i n , dx). If we interpret the updating-selection transition as a birth and death process, then arises the important notion of the ancestral line of a current individual. More precisely, when a particle ξ i n-1 -→ ξ i n evolves to a new location ξ i n , we can interpret ξ i n-1 as the parent of ξ i n . Looking backwards in time and recalling that the particle ξ i n-1 has selected a site ξ j n-1 in the configuration at time (n -1), we can interpret this site ξ j n-1 as the parent of ξ i n-1 and therefore as the ancestor denoted ξ i n-1,n at level (n -1) of ξ i n . Running backwards in time we may trace the whole ancestral line

ξ i 0,n ←-ξ i 1,n ←-. . . ←-ξ i n-1,n ←-ξ i n,n = ξ i n ( 10
)
Most of the terminology we have used is drawn from filtering and genetic evolution theories.

In filtering, the former particle model is dictated by the two steps predictionupdating learning equations of the conditional distributions of a signal process, given some noisy and partial observations. In this setting, the potential functions represent the likelihood function of the current observation, while the free exploration transitions are related to the Markov transitions of the signal process.

In biology, the mutation-selection particle model presented above is used to mimic genetic evolutions of biological organisms and more generally natural evolution processes. For instance, in gene analysis, each population of individuals represents a chromosome and each individual particle is called a gene. In this setting the fitness potential function is usually time-homogeneous and it represents the quality and the adaptation potential value of the set of genes in a chromosome [START_REF] Holland | Adaptation in Natural and Artificial Systems[END_REF]. These particle algorithms are also used in population analysis to model changes in the structure of population in time and in space.

The different types of particle approximation measures associated with the genetic type particle model described above are summarized in the following synthetic picture corresponding to the case N = 3:

• / / • • / / • / / • = • • / / • @ @ @ @ @ @ @ ? ? / / • • / / • = • • / / • • > > } } } } } } } } / / • / / • = •
In the next 4 sections we give an overview of the 4 particle approximation measures can be be extracted from the interacting population evolution model described above. We also provide some basic formulation of the concentration inequalities that will be treated in greater detail later. As a service to the reader we also provide precise pointers to their location within the following chapters. We already mention that the proofs of these results are quite subtle.

The precise form of the constants in these exponential inequalities depends on the contraction properties of Feynman-Kac flows. Our stochastic analysis requires to combine the stability properties of the nonlinear semigroup of the Feynman-Kac distribution flow η n , with deep convergence results of empirical processes theory associated with interacting random samples.

Current population models

The occupation measures of the current population, represented by the red dots in the above figure

η N n := 1 N N i=1 δ ξ i n RR n°7677
converge to the n-th time marginals η n of the Feynman-Kac measures Q n . We shall measure the performance of these particle estimates through several concentration inequalities, with a special emphasis on uniform inequalities w.r.t. the time parameter. Our results will basically be stated as follows.

1) For any time horizon n ≥ 0, any bounded function f , any N ≥ 1, and for any x ≥ 0, the probability of the event

η N n -η n (f ) ≤ c 1 N 1 + x + √ x + c 2 √ N √ x
is greater than 1e -x . In the above display, c 1 stands for a finite constant related to the bias of the particle model, while c 2 is related to the variance of the scheme. The values of c 1 and c 2 don't depend on the time parameter. We already mention one important consequence of these uniform concentration inequalities for time homogeneous Feynman-Kac models. Under some regularity conditions, the flow of measures η n tends to some fixed point distribution η ∞ , in the sense that

η n -η ∞ tv ≤ c 3 e -δn (11)
for some finite positive constants c 3 and δ. The connexions between these limiting measures and the top of the spectrum of Schrödinger operators is discussed in section 2.7.1. We also refer the reader to section 2.7.2 for a discussion on these quasi-invariant measures and Yaglom limits. Quantitative contraction theorems for Feynman-Kac semigroups are developed in the section 3.4.2. As a direct consequence of the above inequalities, we find that for any x ≥ 0, the probability of the following events is is greater than

1 -e -x η N n -η ∞ (f ) ≤ c 1 N 1 + x + √ x + c 2 √ N √ x + c 3 e -δn 2) For any x = (x i ) 1≤i≤d ∈ E n = R d , we set (-∞, x] = d i=1 (-∞,
x i ] and we consider the repartition functions

F n (x) = η n 1 (-∞,x]
and

F N n (x) = η N n 1 (-∞,x]
The probability of the following event

√ N F N n -F n ≤ c d (x + 1)
is greater than 1e -x , for any x ≥ 0, for some universal constant c < ∞ that doesn't depend on the dimension, nor on the time parameter. Furthermore, under the stability properties [START_REF] Cerou | Genealogical Models in Entrance Times Rare Event Analysis Alea[END_REF], if we set

F ∞ (x) = η ∞ 1 (-∞,x]
then, the probability of the following event

F N n -F ∞ ≤ c √ N d (x + 1) + c 3 e -δn
is greater than 1e -x , for any x ≥ 0, for some universal constant c < ∞ that doesn't depend on the dimension.

RR n°7677

For more precise statements, we refer the reader to corollary 6.1, and respectively to corollary 6.5.

The concentration properties of the particle measures η N n around their limiting values are developed in chapter 6. In section 6.3, we design a stochastic perturbation analysis that allows to enter the stability properties of the limiting Feynman-Kac semigroup. Finite marginal models are discussed in section 6.4.1. Section 6.4.2 is concerned with the concentration inequalities of interacting particle processes w.r.t. some collection of functions.

Particle free energy models

Mimicking the multiplicative formula (6), we set and

Z N n = 0≤p<n η N p (G p ) and γ N n (dx) = Z N n × η N n (dx) (12)
We already mention that these rather complex particle models provide an unbiased estimate of the unnormalized measures. That is, we have that

E   η N n (f n ) 0≤p<n η N p (G p )   = E   f n (X n ) 0≤p<n G p (X p )   (13)
The concentration properties of the unbiased particle free energies Z N n around their limiting values Z n are developed in section 6.5. Our results will basically be stated as follows.

For any N ≥ 1, and any ǫ ∈ {+1, -1}, the probability of each of the following events

ǫ n log Z N n Z n ≤ c 1 N 1 + x + √ x + c 2 √ N √ x
is greater than 1e -x . In the above display, c 1 stands for a finite constant related to the bias of the particle model, while c 2 is related to the variance of the scheme. Here again, the values of c 1 and c 2 don't depend on the time parameter. A more precise statement is provided in corollary 6.7.

Genealogical tree model

The occupation measure of the N -genealogical tree model represented by the lines linking the blue dots converges as

N → ∞ to the distribution Q n lim N →∞ 1 N N i=1 δ (ξ i 0,n ,ξ i 1,n ,...,ξ i n,n ) = Q n ( 14
)
Our concentration inequalities will basically be stated as follows. A more precise statement is provided in corollary 6.2.

For any n ≥ 0, any bounded function f n on the path space E n , s.t. f n ≤ 1, and any N ≥ 1, the probability of each of the following events

1 N N i=1 f n (ξ i 0,n , ξ i 1,n , . . . , ξ i n,n ) -Q n (f n ) ≤ c 1 n + 1 N 1 + x + √ x + c 2 (n + 1) N √ x RR n°7677
is greater than 1e -x . In the above display, c 1 stands for a finite constant related to the bias of the particle model, while c 2 is related to the variance of the scheme. Here again, the values of c 1 and c 2 don't depend on the time parameter. The concentration properties of genealogical tree occupation measures can be derived more or less directly from the ones of the current population models. This rather surprising assertion comes from the fact that the n-th time marginal η n of a Feynman-Kac measure associated with a reference historical Markov process has the same form as in the measure [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF]. This equivalence principle between Q n and the marginal measures are developed in section 3.2, dedicated to in historical Feynman-Kac models.

Using these properties, we prove concentration properties for interacting empirical processes associated with genealogical tree models. Our concentration inequalities will basically be stated as follows. A more precise statement is provided in section 6.4.2. We let F n be the set of product functions of indicator of cells in the path space E n = R d0 × . . . , ×R dn , for some d p ≥ 1, p ≥ 0. We also denote by η N n the occupation measure of the genealogical tree model.In this notation, the probability of the following event

sup fn∈Fn η N n (f n ) -Q n (f n ) ≤ c (n + 1) 0≤p≤n d p N (x + 1)
is greater than 1e -x , for any x ≥ 0, for some universal constant c < ∞ that doesn't depend on the dimension.

Complete genealogical tree models

Mimicking the backward model [START_REF] Fr | On the Conditional Distributions of Spatial Point Processes HAL-INRIA RR-7232[END_REF] and the above formulae, we set

Γ N n = Z N n × Q N n (15)
with

Q N n (d(x 0 , . . . , x n )) = η N n (dx n ) n q=1 M q,η N q-1 (x q , dx q-1 )
Notice that the computation of sums w.r.t. these particle measures are reduced to summations over the particles locations ξ i n . It is therefore natural to identify a population of individual (ξ 1 n , . . . , ξ N n ) at time n to the ordered set of indexes {1, . . . , N }. In this case, the occupation measures and the functions are identified with the following line and column vectors

η N n := 1 N , . . . , 1 N and f n :=    f n (ξ 1 n ) . . . f n (ξ N n )   
and the matrices M n,η N n-1 by the (N × N ) matrices

M n,η N n-1 :=    M n,η N n-1 (ξ 1 n , ξ 1 n-1 ) • • • M n,η N n-1 (ξ 1 n , ξ N n-1 ) . . . . . . . . . M n,η N n-1 (ξ N n , ξ 1 n-1 ) • • • M n,η N n-1 (ξ N n , ξ N n-1 )    ( 16
) RR n°7677
with the (i, j)-entries

M n,η N n-1 (ξ i n , ξ j n-1 ) = G n-1 (ξ j n-1 )H n (ξ j n-1 , ξ i n ) N k=1 G n-1 (ξ k n-1 )H n (ξ k n-1 , ξ i n )
For instance, the Q n -integration of normalized additive linear functionals of the form

f n (x 0 , . . . , x n ) = 1 n + 1 0≤p≤n f p (x p ) (17)
is given the particle matrix approximation model

Q N n (f n ) = 1 n + 1 0≤p≤n η N n M n,η N n-1 M 2,η N 1 M p+1,η N p (f p )
These type of additive functionals arise in the calculation of the sensitivity measures discussed in section 2.4.1.

The concentration properties of the particle measures Q N n around the Feynman-Kac measures Q n are developed in section 6.6. Special emphasis is given to the additive functional models [START_REF] Dawson | Large deviations for interacting processes in the strong topology[END_REF]. In section 6.6.3, we extend the stochastic perturbation methodology developed in section 6.3 for time marginal model to the particle backward Markov chain associated with the random stochastic matrices [START_REF] Dawson | Measure-valued Markov processes[END_REF]. This technique allows to enter not only the stability properties of the limiting Feynman-Kac semigroup, but also the ones of the particle backward Markov chain model.

Our concentration inequalities will basically be stated as follows. A more precise statement is provided in corollary 6.9 and in corollary 6.11.

For any n ≥ 0, any normalized additive functional of the form [START_REF] Dawson | Large deviations for interacting processes in the strong topology[END_REF], with max 0≤p≤n f p ≤ 1, and any N ≥ 1, the probability of each of the following events

Q N n -Q n (f n ) ≤ c 1 1 N (1 + (x + √ x)) + c 2 x N (n + 1)
is greater than 1e -x . In the above display, c 1 stands for a finite constant related to the bias of the particle model, while c 2 is related to the variance of the scheme. Here again, the values of c 1 and c 2 don't depend on the time parameter.

For any a = (a i ) 1≤i≤d ∈ E n = R d , we denote by C a the cell

C a := (-∞, a] = d i=1 (-∞, a i ]
and f a,n the additive functional

f a,n (x 0 , . . . , x n ) = 1 n + 1 0≤p≤n 1 (-∞,a] (x p )
The probability of the following event

sup a∈R d Q N n (f a,n ) -Q n (f a,n ) ≤ c d N (x + 1)
is greater than 1e -x , for any x ≥ 0, for some constant c < ∞ that doesn't depend on the dimension, nor on the time horizon.

Remark 1.1 One way to turn all of these inequalities in term of Bernstein style concentration inequalities is as follows. For any exponential inequality of the form

∀x ≥ 0 P X ≤ ax + √ 2bx + c ≤ 1 -e -x
for some non negative constants (a, b, c), we also have

∀y ≥ 0 P (X ≤ y + c) ≤ 1 -exp - y 2 2 (b + ay)
A proof of this result is provided in lemma 4.9.

Basic notation

This section provides some background from stochastic analysis and integral operator theory we require for our proofs. Most of the results with detailed proofs can be located in the book [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], on Feynman-Kac formulae and interacting particle methods. Our proofs also contain cross-references to this well rather known material, so the reader may wish to skip this section and enter directly to the chapter 2 dedicated to some application domains of Feynman-Kac models.

Integral operators

We denote respectively by M(E), M 0 (E), P(E), and B(E), the set of all finite signed measures on some measurable space (E, E), the convex subset of measures with null mass, the set of all probability measures, and the Banach space of all bounded and measurable functions f equipped with the uniform norm f . We also denote by Osc 1 (E), and by B 1 (E) the set of E-measurable functions f with oscillations osc(f ) ≤ 1, and respectively with f ≤ 1. We let

µ(f ) = µ(dx) f (x)
be the Lebesgue integral of a function f ∈ B(E), with respect to a measure µ ∈ M(E).

We recall that the total variation distance on M(E) is defined for any µ ∈ M(E) by

µ tv = 1 2 sup (A,B)∈E 2 (µ(A) -µ(B))
We recall that a bounded integral operator M from a measurable space (E, E) into an auxiliary measurable space (F, F ) is an operator

f → M (f ) from B(F ) into B(E) such that the functions M (f )(x) := F M (x, dy)f (y)
are E-measurable and bounded, for any f ∈ B(F ). A Markov kernel is a positive and bounded integral operator M with M (1) = 1. Given a pair of bounded integral operators (M 1 , M 2 ), we let (M 1 M 2 ) the composition operator defined by

(M 1 M 2 )(f ) = M 1 (M 2 (f )).
For time homogeneous state spaces, we denote by M m = M m-1 M = M M m-1 the m-th composition of a given bounded integral operator M , with m ≥ 1. A bounded integral operator M from a measurable space (E, E) into an auxiliary measurable space (F, F ) also generates a dual operator

µ(dx) → (µM )(dx) = µ(dy)M (y, dx) from M(E) into M(F ) defined by (µM )(f ) := µ(M (f )).
We also used the notation

K [f -K(f )] 2 (x) := K [f -K(f )(x)] 2 (x)
for some bounded integral operator K and some bounded function f . We prefer to avoid unnecessary abstraction and technical assumptions, so we frame the standing assumption that all the test functions are in the unit sphere, and the integral operators, and all the random variables are sufficiently regular that we are justified in computing integral transport equations, regular versions of conditional expectations, and so forth.

Contraction coefficients

When the bounded integral operator M has a constant mass, that is, when M (1) (x) = M (1) (y) for any (x, y) ∈ E 2 , the operator µ → µM maps M 0 (E) into M 0 (F ). In this situation, we let β(M ) be the Dobrushin coefficient of a bounded integral operator M defined by the formula

β(M ) := sup {osc(M (f )) ; f ∈ Osc(F )}
Notice that β(M ) is the operator norm of M on M 0 (E), and we have the equivalent formulations

β(M ) = sup { M (x, .) -M (y, .) tv ; (x, y) ∈ E 2 } = sup µ∈M0(E)
µM tv / µ tv A detailed proof of these well known formulae can be found in [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF].

Given a positive and bounded potential function G on E, we also denote by Ψ G the Boltzmann-Gibbs mapping from P(E) into itself defined for any µ ∈ P(E) by

Ψ G (µ)(dx) = 1 µ(G) G(x) µ(dx)
For ]0, 1]-valued potential functions, we also mention that Ψ G (µ) can be expressed as a non linear Markov transport equation

Ψ G (µ) = µS µ,G (18)
with the Markov transitions

S µ,G (x, dy) = G(x) δ x (dy) + (1 -G(x)) Ψ G (µ)(dy)
We notice that

Ψ G (µ) -Ψ G (ν) = (µ -ν)S µ + ν(S µ -S ν ) and ν(S µ -S ν ) = (1 -ν(G)) [Ψ G (µ) -Ψ G (ν)] RR n°7677
from which we find the formula

Ψ G (µ) -Ψ G (ν) = 1 ν(G) (µ -ν)S µ
In addition, using the fact that

∀(x, A) ∈ (E, E) S µ (x, A) ≥ (1 -G ) Ψ G (µ)(A)
we prove that β(S µ ) ≤ G and

Ψ G (µ) -Ψ G (ν) tv ≤ G µ(G) ∨ ν(G) µ -ν tv
If we set Φ(µ) = Ψ G (µ)M , for some Markov transition M , then we have the decomposition

Φ(µ) -Φ(ν) = 1 ν(G) (µ -ν)S µ M (19)
for any couple of measures ν, µ on E. From the previous discussion, we also find the following Lipschitz estimates

Φ(µ) -Φ(ν) tv ≤ G µ(G) ∨ ν(G) β(M ) µ -ν tv (20)
We end this section with an interesting contraction property of a Markov transition

M G (x, dy) = M (x, dy)G(y) M (G)(x) = Ψ G (δ x M )(dy) (21)
associated with a ]0, 1]-valued potential function G, with

g = sup x,y G(x)/G(y) < ∞ (22)
It is easily checked that

|M G (f )(x) -M G (f )(y)| = |Ψ G (δ x M )(f ) -Ψ G (δ y M )(f )| ≤ g δ x M -δ y M tv from which we conclude that β (M G ) ≤ g β (M ) (23)

Orlicz norms and Gaussian moments

We let π ψ [Y ] be the Orlicz norm of an R-valued random variable Y associated with the convex function ψ(u) = e u 2 -1, and defined by

π ψ (Y ) = inf {a ∈ (0, ∞) : E(ψ(|Y |/a)) ≤ 1} with the convention inf ∅ = ∞. Notice that π ψ (Y ) ≤ c ⇐⇒ E (ψ(Y /c)) ≤ 1 RR n°7677
For instance, the Orlicz norm of a Gaussian and centred random variable U , s.t. E(U 2 ) = 1, is given by π ψ (U ) = 8/3. We also recall that

E U 2m = b(2m) 2m := (2m) m 2 -m E |U | 2m+1 ≤ b(2m + 1) 2m+1 := (2m + 1) (m+1) m + 1/2 2 -(m+1/2) (24)
with (q + p) p := (q + p)!/q!. The second assertion comes from the fact that

E U 2m+1 2 ≤ E U 2m E U 2(m+1)
and therefore

b(2m + 1) 2(2m+1) = E U 2m E U 2(m+1) = 2 -(2m+1) (2m) m (2(m + 1)) (m+1)
This formula is a direct consequence of the following decompositions

(2(m + 1)) (m+1) = (2(m + 1))! (m + 1)! = 2 (2m + 1)! m! = 2 (2m + 1) (m+1)
and

(2m) m = 1 2m + 1 (2m + 1)! m! = 1 2m + 1 (2m + 1) (m+1)
We also mention that b(m) ≤ b(2m)

Indeed, for even numbers m = 2p we have

b(m) 2m = b(2p) 4p = E(U 2p ) 2 ≤ E(U 4p ) = b(4p) 4p = b(2m) 2m
and for odd numbers m = (2p + 1), we have

b(m) 2m = b(2p + 1) 2(2p+1) = E U 2p E U 2(p+1) ≤ E U 2p (2p+1) p p 2p+1 E U 2(p+1) (2p+1) p+1 p+1 2p+1 = E U 2(2p+1) = b(2(2p + 1)) 2(2p+1) = b(2m) 2m
2 Some application domains

Introduction

Feynman-Kac particle methods are also termed quantum Monte Carlo methods in computational physics, genetic algorithms in computer sciences, and particle filters and-or sequential Monte Carlo methods in information theory, as well as in bayesian statistics.

The mathematical foundations of these advanced interacting Monte Carlo methodologies are now fifteen years old [START_REF] Moral | Non Linear Filtering: Interacting Particle Solution[END_REF]. Since this period, so many descriptions and variants of these models have been published in applied probability, signal processing and Bayesian statistic literature. For a detailed discussion on their application domains with a precise bibliography of who first did that when, we refer the reader to any of the following references [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], and [START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF][START_REF] Doucet | Sequential Monte Carlo Methods in Pratice[END_REF].

In the present section, we merely content ourselves in illustrating the rather abstract models (1) with the Feynman-Kac representations of 20 more or less well known conditional distributions, including three more recent applications related to island particle models, functional kinetic parameter derivatives, and gradient analysis of Markov semigroups.

The forthcoming series of examples, combined with their mean field particle interpretation models described in section 1.4, also illustrate the ability of the Feynman-Kac particle methodology to solve complex conditional distribution flows as well as their normalizing constants.

Of course, this selected list of applications does not attempt to be exhaustive. The topics selection is largely influenced by the personal taste of the authors. A complete description on how particle methods are applied in each application model area would of course require separate volumes, with precise computer simulations and comparisons with different types of particle models and other existing algorithms.

We also limit ourselves to describing the key ideas in a simple way, often sacrificing generality. Some applications are nowadays routine, and in this case we provide precise pointers to existing more application-related articles in the literature. Reader who wishes to know more about some specific application of these particle algorithms is invited to consult the referenced papers.

One natural path of "easy reading" will probably be to choose a familiar or attractive application area and to explore some selected parts of the lecture notes in terms of this choice. Nevertheless, this advice must not be taken too literally. To see the impact of particle methods, it is essential to understand the full force of Feynman-Kac modeling techniques on various research domains. Upon doing so, the reader will have a powerful weapon for the discovery of new particle interpretation models. The principal challenge is to understand the theory well enough to reduce them to practice.

Boltzmann-Gibbs measures

Interacting Markov chain Monte Carlo methods

Suppose we are given a sequence of target probability measures on some measurable state space E of the following form

µ n (dx) = 1 Z n    0≤p<n h p (x)    λ(dx) (26)
with some sequence of bounded nonnegative potential functions

h n : x ∈ E → h n (x) ∈ (0, ∞)
and some reference probability measure λ on E. In the above displayed formula, Z n stands for a normalizing constant. We use the convention ∅ = 1 and µ 0 = λ.

We further assume that we have a dedicated Markov chain Monte Carlo transition µ n = µ n M n with prescribed target measures µ n , at any time step. Using the fact that

µ n+1 = Ψ hn (µ n )
with the Boltzmann-Gibbs transformations defined in (5), we prove that

µ n+1 = µ n+1 M n+1 = Ψ hn (µ n )M n+1
from which we conclude that

µ n (f ) = E   f (X n ) 0≤p<n h p (X p )   / E   0≤p<n h p (X p )  
with the reference Markov chain

P (X n ∈ dx | X n-1 ) = M n (X n-1 , dx)
In addition, we have

Z n+1 =   0≤p<n h p (x)   h n (x) µ(dx) = Z n µ n (h n ) = 0≤p≤n µ p (h p )
We illustrate these rather abstract models with two applications related respectively to probability restriction models and stochastic optimization simulated annealing type models. For a more thorough discussion on these interacting MCMC models, and related sequential Monte Carlo methods, we refer the reader to [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo samplers[END_REF].

Probability restrictions

If we choose Markov chain Monte Carlo type local moves

µ n = µ n M n
with some prescribed target Boltzmann-Gibbs measures

µ n (dx) ∝ 1 An (x) λ(dx)
associated with a sequence of decreasing subsets A n ↓, and some reference measure λ, then we find that µ n = η n and Z n = λ(A n ), as soon as the potential functions in ( 1) and ( 26) are chosen so that

G n = h n = 1 An+1
This stochastic model arise in several application domains. In computer science literature, the corresponding particle approximation models are sometimes called subset methods, sequential sampling plans, randomized algorithms, or level splitting algorithms. They were used to solve complex NP-hard combinatorial counting problems [START_REF] Fr | Vaismana Smoothed Splitting Method for Counting[END_REF], extreme quantile probabilities [START_REF] Cerou | Sequential Monte Carlo for Rare Event Estimation[END_REF][START_REF] Guyader | Simulation and Estimation of Extreme Quantiles and Extreme Probabilities[END_REF], and uncertainty propagations in numerical codes [START_REF] Caron | Simulation particulaire, Dimensionnement en conditions extrêmes[END_REF].

Stochastic optimization

If we choose Markov chain Monte Carlo type local moves µ n = µ n M n with some prescribed target Boltzmann-Gibbs measures

µ n (dx) ∝ e -βnV (x) λ(dx)
associated with a sequence of increasing inverse temperature parameters β n ↑, and some reference measure λ, then we find that µ n = η n and Z n = λ(e -βnV ) as soon as the potential functions in ( 1) and ( 26) are chosen so that

G n = h n = e -(βn+1-βn)V
For instance, we can assume that the Markov transition M n = M mn n,βn is the m n -iterate of the following Metropolis Hasting transitions

M n,βn (x, dy) = K n (x, dy) min 1, e -βn(V (y)-V (x)) + 1 -z K n (x, dz) min 1, e -βn(V (z)-V (x)) δ x (dy)
We finish this section with assorted collection of enriching comments on interacting Markov chain Monte Carlo algorithms associated with the Feynman-Kac models described above.

Conventional Markov chain Monte Carlo methods (abbreviated MCMC methods) with time varying target measures µ n can be seen as a single particle model with only mutation explorations according to the Markov transitions M n = K mn n , where K mn n stands for the iteration of an MCMC transition K n s.t. µ n = µ n K n . In this situation, we choose a judicious increasing sequence m n so that the non homogeneous Markov chain is sufficiently stable, even if the target measures become more and more complex to sample. When the target measure is fixed, say of the form µ T for some large T , the MCMC sampler again uses a single particle with behave as a Markov chain with time homogenous transitions M T . The obvious drawback with these two conventional MCMC samplers is that the user does not know how many steps are really needed to be close to the equilibrium target measure. A wrong choice will return samples with a distribution far from the desired target measure.

Interacting MCMC methods run a population of MCMC samplers that interact one each other through a recycling-updating mechanism so that the occupation measure of the current measure converge to the target measure, when we increase the population sizes. In contrast with conventional MCMC methods, there are no burn-in time questions, nor any quantitative analysis to estimate the convergence to equilibrium of the MCMC chain..

Island particle models

In this section, we provide a brief discussion on interacting colonies and island particle models arising in mathematical biology and evolutionary computing literature [START_REF] Whitley | The island model genetic algorithm: On separability, population size and convergence[END_REF][START_REF] Skolicki | The influence of migration sizes and intervals on island models[END_REF]. The evolution of these stochastic island models is again defined in terms of a free evolution and a selection transition. During the free evolution each island evolves separately as a single mean field particle model with mutation-selection mechanism between the individual in the island population. The selection pressure between islands is related to the average fitness of the individuals in island population. A colony with poor fitness is killed, and replaced by brand new generations of "improved" individuals coming from better fitted islands.

For any measurable function f n on E n , we set

X (0) n = X n ∈ E (0) n := E n and f (0) n =
f n and we denote by

X (1) n = X (1,i) n 1≤i≤N1 ∈ E (1) n := E (0) n N1
the N 1 -particle model associated with the reference Markov chain X n defined by

f (1) n (X (1) n ) = 1 N 1 N1 i=1 f (0) n (X (1,i) n )
In this notation, the potential value of the random state X

n is given by the formula

G (1) n (X (1) n ) := 1 N 1 N i=1 G (0) n (X (1,i) n )
By construction, we have the almost sure property

N 1 = 1 =⇒ X (0) n = X (1)
n and G (1) n (X (1) n ) = G (0) n (X (0) n ) More interestingly, by the unbiased properties (13) we have for any population size

N 1 E   f (1) n (X (1) n ) 0≤p<n G (1) p (X (1) p )   = E   f (0) n (X (0) n ) 0≤p<n G (0) p (X (0) p )  
Iterating this construction, we let

X (2) n = X (2,i) n 1≤i≤N2 ∈ E (2) n := E (1) n N2
the N 2 -particle model associated with the reference Markov chain X

n , and the potential function G n defined by

f (2) n (X (2) n ) = 1 N 2 N2 i=1 f (1) n (X (2,i) n )
In this notation, the potential value of the random state X

n is given by the formula

G (2) n (X (2) n ) := 1 N 2 N i=1 G (1) n (X (2,i) n ) RR n°7677
and for any for any population size

N 2 E   f (2) n (X (2) n ) 0≤p<n G (2) p (X (2) p )   = E   f (0) n (X (0) n ) 0≤p<n G (0) p (X (0) p )  

Particle Markov chain Monte Carlo methods

In this section, we present an interacting particle version of the particle Markov chain Monte Carlo method developed in the recent seminal article by C. Andrieu, A. Doucet, and R. Holenstein [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF].

We consider a collection of Markov transition and positive potential functions (M θ,n , G θ,n ) that depend on some random variable Θ = θ, with distribution ν on some state space S. We let η θ,n be the n-time marginal of the Feynman-Kac measures defined as in [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF], by replacing (M n , G n ) by (M θ,n , G θ,n ). We also consider the probability distribution P (θ, dξ) of the N -particle model

ξ := (ξ θ,0 , ξ θ,1 , . . . , ξ θ,T )
on the interval [0, T ], with mutation transitions M θ,n , and potential selection functions G θ,n , with n ≤ T . We fix a large time horizon T , and for any 0 ≤ n ≤ T , we set

µ n (d(ξ, θ)) = 1 Z n    0≤p<n h p (ξ, θ)    λ(d(ξ, θ)) (27)
with some sequence of bounded nonnegative potential functions

h n : (ξ, θ) ∈   0≤p≤T E N p   × S → h n (ξ, θ) ∈ (0, ∞)
the reference measure λ given by

λ(d(ξ, θ)) = ν(dθ) P (θ, dξ)
and some normalizing constants Z n . Firstly, we observe that these target measures have the same form as in [START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF]. Thus, they can be sampled using the Interacting Markov chain Monte Carlo methodology presented in section 2.2.1. Now, we examine the situation where h p is given by the empirical mean value of the potential function G θ,p w.r.t. the occupation measures η N θ,p of the N -particle model ξ θ,p = ξ i θ,p 1≤i≤N associated with the realization Θ = θ; more formally, we have that

h p (ξ, θ) = 1 N 1≤i≤N G θ,p (ξ i θ,p ) = η N θ,p (G θ,p )
Using the unbiased property of the particle free energy models presented in (13), we clearly have

P (θ, dξ)    0≤p<n h p (ξ, θ)    = E   0≤p<n η N θ,p (G θ,p )   = 0≤p<n η θ,p (G θ,p ) RR n°7677
from which we conclude that the Θ-marginal of µ n is given by the following equation

µ n • Θ -1 (dθ) = 1 Z n    0≤p<n η θ,p (G θ,p )    ν(dθ)
We end this section with some comments on these distributions.

As the initiated reader may have certainly noticed, the marginal analysis derived above coincides with one developed in section 2.2.4 dedicated to island particle models.

We also mention that the measures µ n introduced in ( 27) can be approximated using the interacting Markov chain Monte Carlo methodology presented in section 2.2.1, or the particle MCMC methods introduced in the article [START_REF] Andrieu | Particle Markov chain Monte Carlo methods[END_REF].

Last but not least, we observe that

0≤p<n η θ,p (G θ,p ) = E   0≤p<n G θ,p (X θ,p )   = Z n (θ)
where X θ,n stand for the Markov chain with transitions M θ,n , and initial distribution η θ,0 . In the r.h.s. of the above displayed formulae, Z n (θ) stands for the normalizing constant of the Feynman-Kac measures defined as in [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF], by replacing

(M n , G n ) by (M θ,n , G θ,n ). This shows that µ n • Θ -1 (dθ) = 1 Z n Z n (θ) ν(dθ)
The goal of some stochastic optimization problem is to extract the parameter θ that minimizes some mean value functional of the form

θ → Z n (θ) = E   0≤p<n G θ,n (X θ,p )  
For convex functionals, we can use gradient type techniques using the Backward Feynman-Kac derivative interpretation models developed in section 2.4.1 (see also the three joint articles of the first author with A. Doucet, and S. S. Singh [START_REF] Del Moral | A Backward Particle Interpretation of Feynman-Kac Formulae[END_REF][START_REF] Del Moral | Forward Smoothing Using Sequential Monte Carlo Technical Report 638[END_REF][START_REF] Del Moral | Uniform stability of a particle approximation of the optimal filter derivative[END_REF]).

When ν is the uniform measure over some compact set S, an alternative approach is to estimate the measures ( 27) by some empirical measure

1 N 1≤i≤N δ ξ (i) n ,θ (i) n ∈ P     0≤p≤n E N p   × S  
and to select the sampled state

ξ (i) n , θ (i) n := ξ (i,j) 0,n 1≤j≤N , ξ (i,j) 1,n 1≤j≤N , . . . , ξ (i,j) n,n 1≤j≤N , θ (i) n ∈ E N 0 × E N 1 × . . . × E N n × S RR n°7677
that maximizes the empirical objective functional

i ∈ {1, . . . , N } → 0≤p<n 1 N 1≤j≤N G θ (i) n ,p (ξ (i,j) p,n , θ (i) n )

Markov bridges and chains with fixed terminal value

In many applications, it is important to sample paths of Markov chains with prescribed fixed terminal conditions. When the left end starting point is distributed w.r.t. to a given regular probability measure π, we can use the time reversal Feynman-Kac formula presented by the first author and A. Doucet in [START_REF] Del Moral | On a class of genealogical and interacting Metropolis Models[END_REF]. More precisely, for time homogeneous models (G n , M n ) = (G, M ) in transition spaces, if we consider the Metropolis-Hasting ratio

G(x 1 , x 2 ) = π(dx 2 )K(x 2 , dx 1 ) π(dx 1 )M (x 1 , dx 2 )
then we find that

Q n = Law K π ((X 0 , . . . , X n ) | X n = x n )
where Law K π stands for the distribution of the Markov chain starting with an initial condition π and evolving according some Markov transition K. The proof of these formulas are rather technical, we refer the reader the article [START_REF] Del Moral | On a class of genealogical and interacting Metropolis Models[END_REF]n and to the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF].

For initial and terminal fixed end-points, we need to consider the paths distribution of Markov bridges. As we mentioned in the introduction, on page 9, these Markov bridges are particular instances of the reference Markov chains of the abstract Feynman-Kac model [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF]. Depending on the choice of the potential functions in (1), these Markov bridge models can be associated with several application domains, including filtering problems or rare event analysis of bridge processes.

We assume that the elementary Markov transitions M n of the chain X n satisfy the regularity condition [START_REF] Fr | Particle approximations of a class of branching distribution flows arising in multi-target tracking HAL[END_REF] for some density functions H n and some reference measure λ n . In this situation, the semigroup Markov transitions M p,n+1 = M p+1 M p+2 . . . M n+1 are absolutely continuous with respect to the measure λ n+1 , for any 0 ≤ p ≤ n, and we have

M p,n+1 (x p , dx n+1 ) = H p,n+1 (x p , x n+1 ) λ n+1 (dx n+1 )
with the density function

H p,n+1 (x p , x n+1 ) = M p,n (H n+1 (., x n+1 )) (x p )
Thanks to these regularity conditions, we readily check that the paths distribution of Markov bridge starting at x 0 and ending at x n+1 at the final time horizon (n + 1) are given by

B (0,x0),(n+1,xn+1) (d(x 1 , . . . , x n )) := P ((X 1 , . . . , X n ) ∈ d(x 1 , . . . , x n ) | X 0 = x 0 , X n+1 = x n+1 ) = 1≤p≤n M p (x p-1 , dx p ) dMp,n+1(xp,.) dMp-1,n+1(xp-1,.) (x n+1 ) = 1≤p≤n Mp(xp-1,dxp) Hp,n+1(xp,xn+1) Mp(Hp,n+1(.,xn+1))(xp-1)

RR n°7677

Using some abusive Bayesian notation, we can rewrite these formula as follows

p((x 1 , . . . , x n ) | (x 0 , x n+1 )) = p(xn+1|xn) p(xn+1|xn-1) p(x n |x n-1 ) . . . p(xn+1|xp) p(xn+1|xp-1) p(x p |x p-1 ) . . . . . . p(xn+1|x1) p(xn+1|x0) p(x 1 |x 0 ) with dM p (x p-1 , .
)

dλ p (x p ) = p(x p |x p-1 ) and H p,n+1 (x p , x n+1 ) = p(x n+1 |x p )
For linear-Gaussian models, the Markov bridge transitions

M p (x p-1 , dx p ) H p,n+1 (x p , x n+1 ) M p (H p,n+1 (., x n+1 )) (x p-1 ) = p(x n+1 |x p ) p(x n+1 |x p-1 ) p(x p |x p-1 )λ p (dx p )
can be explicitly computed using the traditional regression formula, or equivalently the updating step of the Kalman filter.

Rare event analysis

Importance sampling and twisted measures

Computing the probability of some events of the form {V n (X n ) ≥ a}, for some energy like function V n and some threshold a is often performed using the importance sampling distribution of the state variable X n with some multiplicative Boltzmann weight function e βVn(Xn) associated with some temperature parameter β. These twisted measures can be described by a Feynman-Kac model in transition space by setting

G n (X n-1 , X n ) = e β[Vn(Xn)-Vn-1(Xn-1)]
For instance, it is easily checked that

P (V n (X n ) ≥ a) = E f n (X n ) e Vn(Xn) = E   f n (X n ) 0≤p<n G p (X p )   with X n = (X n , X n+1 ) and G n (X n ) = e Vn+1(Xn+1)-Vn(Xn)
and the test function

f n (X n ) = 1 Vn(Xn)≥a e -Vn(Xn)
In the same vein, we have

E (ϕ n (X 0 , . . . , X n ) | V n (X n ) ≥ a) = E F n,ϕn (X 0 , . . . , X n ) e Vn(Xn) /E F n,1 (X 0 , . . . , X n ) e Vn(Xn) = Q n (F n,ϕn )/Q n (F n,1 ) RR n°7677
with the function

F n,ϕn (X 0 , . . . , X n ) = ϕ n (X 0 , . . . , X n ) 1 Vn(Xn)≥a e -Vn(Xn)
We illustrate these rather abstract formula with a Feynman-Kac formulation of European style call options with exercise price a at time n. The prices of these financial contracts are given by formulae of the following form

E ((V n (X n ) -a) + ) = E (V n (X n ) -a) 1 Vn(Xn)≥a = P (V n (X n ) ≥ a) × E ((V n (X n ) -a) | V n (X n ) ≥ a )
It is now a simple exercise to check that these formulae fit with the Feynman-Kac importance sampling model discussed above. Further details on these models, including applications in fiber optics communication and financial risk analysis can also be found in the couple of articles [START_REF] Del Moral | Genealogical Particle Analysis of Rare events[END_REF][START_REF] Del Moral | Simulations of rare events in fiber optics by interacting particle systems[END_REF] and in the article [START_REF] Carmona | Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses[END_REF].

Rare event excursion models

If we consider Markov excursion type models between a sequence of decreasing subsets A n or hitting a absorbing level B, then choosing an indicator potential function that detects if the n-th excursion hits A n before B we find that

Q n = Law( X hits A n | X hits A n before B)
and

P(X hits A n before B) = E   0≤p≤n 1 Ap (X Tp )   = E   0≤p<n G p (X p )  
with the random times

T n := inf {p ≥ T n-1 : X p ∈ (A n ∪ B)}
and the excursion models

X n = (X p ) p∈[Tn,Tn+1] & G n (X n ) = 1 An+1 (X Tn+1 )
In this notation, it is also easily checked that

E f n (X [0,Tn+1] ) | X hits A n before B = Q n (f n )
For a more thorough discussion on these excursion particle model, we refer the reader to the series of articles [START_REF] Cerou | Genealogical Models in Entrance Times Rare Event Analysis Alea[END_REF][START_REF] Cerou | Topart Some recent improvements to importance splitting[END_REF][START_REF] Del Moral | Branching and interacting particle interpretation of rare event probabilities[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Johansen | Sequential Monte Carlo Samplers for Rare Events[END_REF].

Sensitivity measures

Kinetic sensitivity measures

We let θ ∈ R d be some parameter that may represent some kinetic type parameters related free evolution model or to the adaptive potential functions.

We assume that the free evolution model X (θ) k associated to some value of the parameter θ, is given by a one-step probability transition of the form

M (θ) k (x, dx ′ ) := P X (θ) k ∈ dx ′ |X (θ) k-1 = x = H (θ) k (x, x ′ ) λ k (dx ′ )
for some positive density functions H (θ) k and some reference measures λ k . We also consider a collection of functions

G (θ) k = e -V (θ)
k that depend on θ. We also assume that the gradient and the Hessian of the logarithms of these functions w.r.t. the parameter θ are well defined. We let Γ θ n be the Feynman-Kac measure associated with a given value of θ defined for any function f n on the path space E n by

Γ θ n (f n ) = E   f n (X (θ) 0 , . . . , X (θ) n ) 0≤p<n G (θ) p X (θ) p   (28)
We denote by Γ (θ,N ) n

the N -particle approximation measures associated with a given value of the parameter θ and defined in [START_REF] Dawson | Stochastic evolution equations and related measure processes[END_REF].

By using simple derivation calculations, we prove that the first order derivative of the option value w.r.t. θ is given by

∇Γ (θ) n (f n ) = Γ (θ) n (f n Λ (θ) n ) ∇ 2 Γ (θ) n (f n ) = Γ (θ) n f n (∇L (θ) n ) ′ (∇L (θ) n ) + f n ∇ 2 L (θ) n with Λ (θ) n := ∇L (θ)
n and the additive functional

L (θ) n (x 0 , . . . , x n ) := n p=1 log G (θ) p-1 (x p-1 )H (θ) p (x p-1 , x p )
These quantities are approximated by the unbiased particle models

∇ N Γ (θ) n (f n ) := Γ (θ,N ) n (f n Λ (θ) n ) ∇ 2 N Γ (θ) n (f n ) = Γ (θ,N ) n f n (∇L (θ) n ) ′ (∇L (θ) n ) + f n ∇ 2 L (θ) n
For a more thorough discussion on these Backward Feynman-Kac models, we refer the reader to the three joint articles of the first author with A. Doucet, and S. S. Singh [START_REF] Del Moral | A Backward Particle Interpretation of Feynman-Kac Formulae[END_REF][START_REF] Del Moral | Forward Smoothing Using Sequential Monte Carlo Technical Report 638[END_REF][START_REF] Del Moral | Uniform stability of a particle approximation of the optimal filter derivative[END_REF].

Gradient estimation of Markov semigroup

We assume that the underlying stochastic evolution is given by an iterated R dvalued random process given by the following equation

X n+1 := F n (X n ) = (F n • F n-1 • • • • • F 0 )(X 0 ) (29)
starting at some random state X 0 , with a sequence of random smooth functions of the form

F n (x) = F n (x, W n ) (30)
with some smooth collection of functions

F n : (x, w) ∈ R d+d ′ → F n (x, w) ∈ R d
and some collection of independent, and independent of s, random variables W n taking values in some R d ′ , with d ′ ≥ 1. The semigroup of the Markov chain X n is the expectation operator defined for any regular function f n and any state x by

P n+1 (f n+1 )(x) := E (f n+1 (X n+1 ) | X 0 = x) = E (f (X n+1 (x)))
with the random flows (X n (x)) n≥0 defined for any n ≥ 0 by the following equation

X n+1 (x) = F n (X n (x))
with the initial condition X 0 (x) = x. By construction, for any 1 ≤ i, j ≤ d and any x ∈ R d we have the first variational equation

∂X i n+1 ∂x j (x) = 1≤k≤d ∂F i n ∂x k (X n (x)) ∂X k n ∂x j (x) (31)
This clearly implies that

∂P n+1 (f ) ∂x j (x) = E   1≤i≤d ∂f ∂x i (X n+1 (x)) ∂X i n+1 ∂x j (x)   (32)
We denote by

V n = (V (i,j) n ) 1≤i,j≤d and A n = (A (i,j) n
) 1≤i,j≤d the random (d × d) matrices with the i-th line and j-th column entries

V (i,j) n (x) = ∂S i n ∂x j (x)
and

A (i,j) n (x) = ∂F i n ∂x j (x) = ∂F i n (., W n ) ∂x j (x) := A (i,j) n (x, W n )
In this notation, the equation ( 31) can be rewritten in terms of the following random matrix formulae

V n+1 (x) = A n (X n (x)) V n (x) := n p=0 A p (X p (x)) (33)
with a product n p=0 A p of noncommutative random elements A p taken in the order A n , A n-1 ,. . . , A 0 . In the same way, the equation ( 32) can be rewritten as

∇P n+1 (f n+1 )(x) = E (∇f n+1 (X n+1 ) V n+1 | X 0 = x) (34)
with

V n+1 := 0≤p≤n A p (X p )
We equip the space R d with some norm . . We assume that for any state U 0 in the unit sphere S d-1 , we have

V n+1 U 0 > 0 RR n°7677
In this situation, we have the multiplicative formulae

∇f n+1 (X n+1 ) V n+1 U 0 = [∇f n+1 (X n+1 ) U n+1 ] 0≤p≤n A p (X p ) U p
with the well defined S d-1 -valued Markov chain defined by

U n+1 = A n (X n )U n / A n (X n )U n ⇔ U n+1 = V n+1 U 0 V n+1 U 0
If we choose U 0 = u 0 , then we obtain the following Feynman-Kac interpretation of the gradient of a semigroup

∇P n+1 (f n+1 )(x) u 0 = E   F n+1 (X n+1 ) 0≤p≤n G p (X p )   (35)
In the above display, X n is the Markov chain sequence

X n := (X n , U n , W n )
starting at (x, u 0 , W 0 ), and the functions F n+1 and G n are defined by

F n+1 (x, u, w) := ∇f n+1 (x) u and G n (x, u, w) := A n (x, w) u
In computational physics literature, the mean particle approximations of these non commutative Feynman-Kac models are often referred as Resampled Monte Carlo methods [START_REF] Vanneste | Estimating generalized Lyapunov exponents for products of random matrices[END_REF].

Roughly speaking, besides the fact that formula [START_REF] Del Moral | Simulations of rare events in fiber optics by interacting particle systems[END_REF] provides a explicit functional Feynman-Kac description of the the gradient of a Markov semigroup, the random evolution model U n on the unite sphere may be degenerate. More precisely, the Markov chain X n = (X n , U n , W n ) may not satisfy the regularity properties stated in section 3.4.1. We end this section with some rather crude upper bound that can be estimated uniformly w.r.t. the time parameter under appropriate regularity conditions on the reduced Markov chain model (X n , W n ). Firstly, we notice that

G n (x, u, w) := A n (x, w) u ≤ G n (x, w) := A n (x, w) := sup u∈S d-1 A n (x, w) u
This implies that

∇P n+1 (f n+1 )(x) := sup 1≤i≤d ∂ ∂x i P n+1 (f n+1 )(x) ≤ F n+1 × E   0≤p≤n G p (X p , W p )  
The r.h.s. functional expectation in the above equation can be approximated using the particle approximation [START_REF] Cerou | Topart Some recent improvements to importance splitting[END_REF] of the multiplicative Feyman-Kac formulae (6), with reference Markov chain (X n , W n ) and potential functions G n .

2.5 Partial observation models

Nonlinear filtering models

In this section we introduce one of the most important example of estimation problem with partial observation, namely the nonlinear filtering model. This model has been the starting point of the application of particle models to engineering sciences, and more particularly to advanced signal processing.

The first rigorous subject on the stochastic modeling, and the rigorous theoretical analysis of particle filters has been started in the mid 1990's in the article [START_REF] Moral | Non Linear Filtering: Interacting Particle Solution[END_REF]. For a detailed discussion on the application domains of particle filtering, with a precise bibliography we refer the reader to any of the following references [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF], and [START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF][START_REF] Doucet | Sequential Monte Carlo Methods in Pratice[END_REF].

The typical model is given by a reference Markov chain model X n , and some partial and noisy observation Y n . The pair process (X n , Y n ) usually forms a Markov chain on some product space E X × E Y with elementary transitions given

P ((X n , Y n ) ∈ d(x, y) | (X n-1 , Y n-1 )) = M n (X n-1 , dx) g n (x, y) λ n (dy) (36)
for some positive likelihood function g n , and some reference probability measure λ n on E Y , and the elementary Markov transitions M n of the Markov chain X n .

If we take

G n (x n ) = p n (y n |x n ) = g n (x n , y n ) (37)
the likelihood function of a given observation Y n = y n and a signal state X n = x n associated with a filtering or an hidden Markov chain problem, then we find that

Q n = Law((X 0 , . . . , X n ) | ∀0 ≤ p < n Y p = y p )
and Z n+1 = p n (y 0 , . . . , y n )

In this context, the optimal one step predictor η n and the optimal filter η n are given by the n-th time marginal distribution

η [y0,...,yn-1] n = η n = Law (X n | ∀0 ≤ p < n Y p = y p ) (38)
and

η [y0,...,yn] n = η n = Ψ Gn (η n ) = Law (X n | ∀0 ≤ p ≤ n Y p = y p ) (39)
Remark 2.1 We can combine these filtering models with the probability restriction models discussed in section 2.2.2, or with the rare event analysis presented in section 2.3. For instance, if we replace the potential likelihood function G n defined in [START_REF] Del Moral | A central limit theorem for nonlinear filtering using interacting particle systems[END_REF] by the function

G n (x n ) = g n (x n , y n ) 1 An (x n )
then we find that

Q n = Law((X 0 , . . . , X n ) | ∀0 ≤ p < n Y p = y p , X p ∈ A p ) RR n°7677

Approximated filtering models

We return to the stochastic filtering model discussed in section 2.5.1. In some instance, the likelihood functions x n → g n (x n , y n ) in ( 37) are computationally intractable, or too expensive to evaluate.

To solve this problem, a natural solution is to sample pseudo-observations. The central idea is to sample the signal-observation Markov chain

X n = (X n , Y n ) ∈ E X = (E X × E Y )
and compare the values of the sampled observations with the real observations.

To describe with some precision these models, we notice that the transitions of X n are given by

M n (X n-1 , d(x, y)) = M n (X n-1 , dx) g n (x, y) λ n (dy)
To simplify the presentation, we further assume that E Y = R d , for some d ≥ 1, and we let g be a Borel bounded non negative function such that

g(u)du = 1 ug(u) du = 0 and |u| 3 g(u) du < ∞
Then, we set for any ǫ > 0, and any

x = (x, y) ∈ (E X × E Y ) g ǫ,n ((x, y), z) = ǫ -d g ((y -z)/ǫ)
Finally, we let (X n , Y ǫ n ) be the Markov chain on the augmented state space

E X × E Y = (E X × E Y ) × E Y with transitions given P (X n , Y ǫ n ) ∈ d(x, y) | (X n-1 , Y ǫ n-1 ) = M n (X n-1 , dx) g ǫ,n (x, y) dy (40)
This approximated filtering problem has exactly the same form as the one introduced in [START_REF] Del Moral | Large deviations for interacting particle systems. Applications to nonlinear filtering problems[END_REF]. Here, the particle approximation model are defined in terms of signal-observation valued particles, and the selection potential function is given by the pseudo-likelihood functions g ǫ,n (., y n ), where y n stands for the value of the observation sequence at time n.

For a more detailed discussion on these particle models, including the convergence analysis of the approximated filtering model, we refer the reader to the article [START_REF] Del Moral | The Monte-Carlo Method for filtering with discrete-time observations[END_REF][START_REF] Del Moral | The Monte-Carlo method for filtering with discrete-time observations: Central limit theorems[END_REF]. These particle models are sometimes called convolution particle filters [START_REF] Vila | Nonlinear filtering in discrete time: A particle convolution approach, Biostatistic group of Montpellier[END_REF]. In Bayesian literature, these approximated filtering models are termed as Approximate Bayesian Computation (and often abbreviated with the acronym ABC).

Parameter estimation in hidden Markov chain models

We consider a pair signal-observation filtering model (X, Y ) that depend on some random variable Θ with distribution µ on some state space S. Arguing as above, if we take

G θ,n (x n ) = p n (y n |x n , θ)
the likelihood function of a given observation Y n = y n and a signal state X n = x n and a realization of the parameter Θ = θ, then the n-th time marginal of Q n is given by

η θ,n = Law(X n | ∀0 ≤ p < n Y p = y p , θ)
Using that the multiplicative formula (6), we prove that

Z n+1 (θ) = p n (y 0 , . . . , y n |θ) = 0≤p≤n η θ,p (G θ,p ) with η θ,p (G θ,p ) = p(y p |y 0 , . . . , y p-1 , θ) = p(y p |x p , θ) dp(x p |θ, y 0 , . . . , y p-1 ) = G θ,p (x p ) η θ,p (dx p )
from which we conclude that

P(Θ ∈ dθ | ∀0 ≤ p ≤ n Y p = y p ) = 1 Z n Z n (θ) µ(dθ)
with

Z n := Z n (θ) µ(dθ)
In some instance, such as in conditionally linear Gaussian models, the normalizing constants Z n (θ) can be computed explicitly, and we can use a Metropolis-Hasting style Markov chain Monte Carlo method to sample the target measures µ n . As in section 2.2.1, we can also turn this scheme into an interacting Markov chain Monte Carlo algorithm.

Indeed, let us choose a Markov chain Monte Carlo type local moves µ n = µ n M n with prescribed target measures

µ n (dθ) := 1 Z n Z n (θ) µ(dθ) Notice that Z n+1 (θ) = Z n (θ) × η θ,n (G θ,n ) ⇒ µ n+1 = Ψ Gn (µ n )
with the Boltzmann-Gibbs transformations defined in (5) associated with the potential function

G n (θ) = η θ,n (G θ,n )
By construction, we have

µ n+1 = µ n+1 M n+1 = Ψ Gn (µ n )M n+1
from which we conclude that

µ n (f ) = E   f (θ n ) 0≤p<n G p (θ p )   / E   0≤p<n G p (θ p )   RR n°7677
with the reference Markov chain

P (θ n ∈ dθ|θ n-1 ) = M n (θ n-1 , dθ)
In addition, we have

Z n+1 = Z n (θ) G n (θ) µ(dθ) = Z n µ n (G n ) = 0≤p≤n µ p (G p )
Remark 2.2 For more general models, we can use the particle Markov chain Monte Carlo methodology presented in section 2.2.5. When the likelihood functions are too expensive to evaluate, we can also combine these particle models with the pseudo-likelihood stochastic models (40) discussed in section 2.5.2.

Interacting Kalman-Bucy filters

We use the same notation as above, but we assume that Θ = (Θ n ) n≥0 is a random sample of a stochastic process Θ n taking values in some state spaces S n . If we consider the Feynman-Kac model associated with the Markov chain

X n = (Θ n , η Θ,n
) and the potential functions

G n (X n ) = η Θ,n (G Θ,n )
then we find that

Q n = Law(Θ 0 , . . . , Θ n | ∀0 ≤ p < n Y p = y p )
and the n-th time marginal are clearly given by

η n = Law(Θ n | ∀0 ≤ p < n Y p = y p )
Assuming that the pair (X, Y ) is a linear and gaussian filtering model given Θ, the measures η Θ,n coincide with the one step predictor of the Kalman-Bucy filter and the potential functions G n (X n ) can be easily computed by gaussian integral calculations. In this situation, the conditional distribution of the parameter Θ is given by a Feynman-Kac model Q n of a the free Markov chain X n weighted by some Boltzmann-Gibbs exponential weight function 0≤p<n G p (X p ) = p n (y 0 , . . . , y n |Θ 0 , . . . , Θ n ) that reflects the likelihood of the path sequence (Θ 0 , . . . , Θ n ). For a more thorough discussion on these interacting Kalman filters, we refer the reader to section 2.6 and section 12.6 in the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF].

Multi-target tracking models

Multiple-target tracking problems deal with correctly estimating several manoeuvring and interacting targets simultaneously given a sequence of noisy and partial observations. At every time n, the first moment of the occupation measure X n := Nn i=1 δ X i n of some spatial branching signal is given for any regular function f by the following formula:

γ n (f ) := E (X n (f )) with X n (f ) := f (x) X n (dx)
For null spontaneous birth measures, these measures coincide with that of an unnormalized Feynman-Kac model with some spatial branching potential functions G n and some free evolution target model X n .

In more general situations, the approximate filtering equation is given by the Malher's multi-objective filtering approximation based on the propagation of the first conditional moments of Poisson approximation models [START_REF] Mahler | A theoretical foundation for the Stein-Winter Probability Hypothesis Density (PHD) multi-target tracking approach[END_REF][START_REF] Mahler | Multi-target Bayes filtering via first-order multi-target moments[END_REF]. These evolution equations are rather complex to introduce and notationally consuming. Nevertheless, as the first moment evolution of any spatial and marked branching process, they can be abstracted by an unnormalized Feynman-Kac model with nonlinear potential functions [START_REF] Fr | Particle approximations of a class of branching distribution flows arising in multi-target tracking HAL[END_REF][START_REF] Fr | On the Conditional Distributions of Spatial Point Processes HAL-INRIA RR-7232[END_REF][START_REF] Fr | On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering HAL[END_REF].

Optimal stopping problems with partial observations

We consider the partially observed Markov chain model discussed in [START_REF] Del Moral | Large deviations for interacting particle systems. Applications to nonlinear filtering problems[END_REF]. The Snell envelop associated with an optimal stopping problem with finite horizon, payoff style function f n (X n , Y n ), and noisy observations Y n as some Markov process, is given by

U k := sup τ ∈T Y k E(f τ (X τ , Y τ )|(Y 0 , . . . , Y k ))
where T Y k stands for the set of all stopping times τ taking values in {k, . . . , n}, whose values are measurable w.r.t. the sigma field generated by the observation sequence Y p , from p = 0 up to the current time k. We denote by η [y0,...,yn-1] n and η [y0,...,yn] n the conditional distributions defined in [START_REF] Del Moral | On the stability of Measure Valued Processes with Applications to filtering[END_REF] and [START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF]. In this notation, for any 0 ≤ k ≤ n we have that

E(f τ (X τ , Y τ )|(Y 0 , . . . , Y k )) = E F τ Y τ , η [Y0,...,Yτ ] τ | (Y 0 , . . . , Y k ) (41)
with the conditional payoff function

F p Y p , η [Y0,...,Yp] p = η [Y0,...,Yp] p (dx) f p (X p , Y p )
It is rather well known that

X p := Y p , η [Y0,...,Yp]
p is a Markov chain with elementary transitions defined by

E F p Y p , η [Y0,...,Yp] p Y p-1 , η [Y0,...,Yp-1] p-1 = (y, µ)
= λ p (dy p ) µM p (g p (., y p )) F p y p , Ψ gp(.,yp) (µM p )

A detailed proof of this assertion can be found in any textbook on advanced stochastic filtering theory. For instance, the book of W. Runggaldier, L. Stettner [START_REF] Runggaldier | Approximations of Discrete Time Partially Observed Control Problems Applied Mathematics Monographs CNR[END_REF] provides a detailed treatment on discrete time non linear filtering, and related partially observed control models.

Roughly speaking, using some abusive Bayesian notation, we have From the above discussion, we can rewrite [START_REF] Del Moral | The Monte-Carlo method for filtering with discrete-time observations: Central limit theorems[END_REF] as the Snell envelop of a fully observed augmented Markov chain sequence

E(f τ (X τ , Y τ )|(Y 0 , . . . , Y k )) = E (F τ (X τ ) | (X 0 , . . . , X k ))
The Markov chain X n takes values in an infinite dimensional state space, and it can rarely be sampled without some addition level of approximation. Using the N -particle approximation models, we can replace the chain X n by the N -particle approximation model defined by

X N n := Y p , η ([Y0,...,Yp],N ) p where η ([Y0,...,Yp],N ) p := Ψ gp(.,Yp) η ([Y0,...,Yp-1,N )] p-1
stands for the updated measure associated associated with the likelihood selection functions g p (., Y p ). The N -particle approximation of the Snell envelop is now given by

E(f τ (X τ , Y τ )|(Y 0 , . . . , Y k )) ≃ N ↑∞ E F τ X N τ | (X N 0 , . . . , X N k )
In this interpretation, the N -approximated optimal stopping problem amounts to compute the quantities

U N k := sup τ ∈T N k E F τ X N τ | (X N 0 , . . . , X N k )
where T N k stands for the set of all stopping times τ taking values in {k, . . . , n}, whose values are measurable w.r.t. the sigma field generated by the Markov chain sequence X N k , from p = 0 up to time k.

Markov chain restriction models

2.6.1 Markov confinement models One of the simplest example of Feynman-Kac conditional distributions is given by choosing indicator functions G n = 1 An of measurable subsets

A n ∈ E n s.t. P (∀0 ≤ p < n X p ∈ A p ) > 0.
In this situation, it is readily checked that

Q n = Law((X 0 , . . . , X n ) | ∀0 ≤ p < n X p ∈ A p )
and

Z n = P (∀0 ≤ p < n X p ∈ A p )
This Markov chain restriction model fits into the particle absorption model (3) presented in the introduction. For a detailed analysis of these stochastic models, and their particle approximations, we refer the reader to the articles [START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF][START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Del Moral | Particle Motions in Absorbing Medium with Hard and Soft Obstacles[END_REF], and the monograph [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF].

Directed polymers and self-avoiding walks

The conformation of polymers in a chemical solvent can be seen as the realization of a Feynman-Kac distribution of a free Markov chain weighted by some Boltzmann-Gibbs exponential weight function that reflects the attraction or the repulsion forces between the monomers. For instance, if we consider the historical process

X n = (X 0 , . . . , X n ) and G n (X n ) = 1 ∈{Xp, p<n} (X n ) then we find that Q n = Law(X n | ∀0 ≤ p < n X p ∈ A p ) = Law((X 0 , . . . , X n ) | ∀0 ≤ p < q < n X p = X q )
with the set A n = {G n = 1}, and the normalizing constants

Z n = P(∀0 ≤ p < q < n X p = X q )

Particle absorption models

We return to the particle absorption model (3) presented in the introduction.

For instance, we can assume that the potential function G n and Markov transitions M n are defined by G n (x) = e -Vn(x)h , and

M n (x, dy) = (1 -λ n h) δ x (dy) + λ n h K n (x, dy) (42)
for some non negative and bounded function V n , some positive parameter λ n ≤ 1/h, h > 0, and some Markov transition K n . We also mention that the confinement models described above can also be interpreted as a particle absorption model related to hard obstacles. In branching processes and population dynamics literature, the model X c n often represent the number of individuals of a given specie [START_REF] Ferrari | Existence of quasistationary distributions. A renewal dynamical approach[END_REF][START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF][START_REF] Steinsaltz | Markov mortality models: implications of quasistationarity and varying initial distributions[END_REF]. Each individual can die or reproduce. The state 0 ∈ E n = N is interpreted as a trap, or as an hard obstacle, in the sense that the specie disappear as soon as X c n hits 0. For a more thorough discussion on particle motions in absorbing medium with hard and soft obstacles, we refer the reader to the pair of articles [START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF][START_REF] Del Moral | Particle Motions in Absorbing Medium with Hard and Soft Obstacles[END_REF].

Doob h-processes

We consider a time homogeneous Feynman-Kac model (G n , M n ) = (G, M ) on some measurable state space E, and we set

Q(x, dy) = G(x)M (x, dy)
We also assume that G is uniformly bounded above and below by some positive constant, and the Markov transition M is reversible w.r.t. some probability measure µ on E, with M (x, .) ≃ µ and dM (x, .)/dµ ∈ L 2 (µ). We denote by λ the largest eigenvalue of the integral operator Q on L 2 , and by h(x) a positive eigenvector

Q(h) = λh
The Doob h-process corresponding to the ground state eigenfunction h defined above is a Markov chain X h n with the time homogeneous Markov transition

M h (x, dy) := 1 λ × h -1 (x)Q(x, dy)h(y) = M (x, dy)h(y) M (h)(x)
and initial distribution η h 0 (dx) ∝ h(x) η 0 (dx). By construction, we have G = λh/M (h) and therefore

Γ n (d(x 0 , . . . , x n )) = λ n η 0 (h) P h n (d(x 0 , . . . , x n )) 1 h(x n )
where P h n stands for the law of the historical process

X h n = (X h 0 , . . . , X h n ) We conclude that dQ n = 1 E(h -1 (X h n )) h -1 (X h n ) dP h n
with the normalizing constants

Z n = λ n η 0 (h) E(h -1 (X h n ))

Yaglom limits and quasi-invariant measures

We return to the time homogeneous Feynman-Kac models introduced in section 2.7.1. Using the particle absorption interpretation (3) we have

Law((X c 0 , . . . , X c n ) | T c ≥ n) = 1 E(h -1 (X h n )) h -1 (X h n ) dP h n and Z n = P (T c ≥ n) = λ n η 0 (h) E(h -1 (X h n )) -→ n↑∞ 0 (43)
Letting η h n := Law(X h n ), we readily prove the following formulae

η n = Ψ 1/h (η h n ) and η h n = Ψ h (η n )
Whenever it exists, the Yaglom limit of the measure η 0 is is defined as the limiting of measure

η n -→ n↑∞ η ∞ = Ψ G (η ∞ )M (44)
of the Feynman-Kac flow η n , when n tends to infinity. We also say that η 0 is quasi-invariant measure is we have η 0 = η n , for any time step. When the Feynman-Kac flow η n is asymptotically stable, in the sense that it forgets its initial conditions, we also say that the quasi-invariant measure η ∞ is the Yaglom measure. Whenever it exist, we let η h ∞ be the invariant measure of the h-process X h n . Under our assumptions, it is a now simple exercise to check that

η ∞ = Ψ M(h) (µ) and η h ∞ := Ψ h (η ∞ ) = Ψ hM(h) (µ)
Quantitative convergence estimates of the limiting formulae ( 43) and ( 44) can be derived using the stability properties of the Feynman-Kac models developed in chapter 3. For a more thorough discussion on these particle absorption models, we refer the reader to the articles of the first author with A. Guionnet [START_REF] Del Moral | On the stability of Measure Valued Processes with Applications to filtering[END_REF][START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF], the ones with L. Miclo [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF], the one with A. Doucet [START_REF] Del Moral | Particle Motions in Absorbing Medium with Hard and Soft Obstacles[END_REF], and the monographs [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF].

3 Feynman-Kac semigroup analysis

Introduction

As we mentioned in section 1.4, the concentration analysis of particle models is intimately related to the regularity properties of the limiting nonlinear semigroup. In this short section, we survey some selected topics on the theory of Feynman-Kac semigroup developed in the series of articles [START_REF] Del Moral | On the stability of interacting processes with applications to filtering and genetic algorithms[END_REF][START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Del Moral | A Backward Particle Interpretation of Feynman-Kac Formulae[END_REF]. For more recent treatments, we also refer the reader to the books [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF].

We begin this chapter with a discussion on path space models. Section 3.2 is concerned with Feynman-Kac historical processes and Backward Markov chain interpretation models We show that the the n-th marginal measures η n of Feynman-Kac model with a reference historical Markov process coincides with the path space measures Q n introduced in (1).

The second part of this section is dedicated to the proof of the Backward Markov chain formulae [START_REF] Fr | On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering HAL[END_REF]. In section 3.3, we analyze the regularity and the semigroup structure of the normalized and unnormalized Feynman-Kac distribution flows η n and γ n .

Section 3.4 is concerned with the stability properties of the normalized Feynman-Kac distribution flow. In a first section, section 3.4.1, we present regularity conditions on the potential functions G n and on the Markov transitions M n , under which the Feynman-Kac semigroup forgets exponentially fast its initial condition. Quantitative contraction theorems are provided in section 3.4.2.

We illustrate these results with three applications related respectively to time discretization techniques, simulated annealing type schemes, and path space models.

The last two sections of this chapter, section 3.5 and section 3.6, are concerned with mean field stochastic particle models and local sampling random field models.

Historical and backward models

The historical process associated with some reference Markov chain X n is defined by the sequence of random paths

X n = (X 0 , . . . , X n ) ∈ E n := (E 0 × . . . × E n )
Notice that the Markov transitions of the chain X n is given for any x n-1 = (x 0 , . . . , x n-1 ) and y n = (y 0 , . . . , y n ) = (y n-1 , y n ) by the following formulae

M n (x n-1 , dy n ) = δ xn-1 (dy n-1 ) M n (y n-1 , dx n ) (45)
We consider a sequence of (0, 1]-valued potential functions G n on E n whose values only depend on the final state of the paths; that is, we have that

G n : x n = (x 0 , . . . , x n ) ∈ E n → G n (x n ) = G n (x n ) ∈ (0, 1] (46)
with some (0, 1]-valued potential function G n on E n . We let (γ n , η n ) the Feynman-Kac model associated with the pair (G n , M n ) on the path spaces E n . By construction, for any function f n on E n , we have

γ n (f n ) = E   f n (X n ) 0≤p<n G p (X p )   = E   f n (X 0 , . . . , X n ) 0≤p<n G p (X p )  
from which we conclude that

γ n = Z n Q n and η n = Q n ( 47
)
where Q n is the Feynman-Kac measure on path space associated with the pair (G n , M n ), and defined in [START_REF] Adamczak | A tail inequality for suprema of unbounded empirical processes with applications to Markov chains[END_REF]. We end this section with the proof of the backward formula [START_REF] Fr | On the Stability and the Approximation of Branching Distribution Flows, with Applications to Nonlinear Multiple Target Filtering HAL[END_REF]. Using the decomposition

Q n (d(x 0 , . . . , x n )) = Z n-1 Z n Q n-1 (d(x 0 , . . . , x n-1 )) Q n (x n-1 , dx n )
we prove the following formulae

η n (dx n ) = Z n-1 Z n η n-1 Q n (dx n ) (48) = Z n-1 Z n η n-1 (G n-1 H n (., x n )) λ n (dx n ) (49)
and

Z n Z n-1 = η n-1 Q n (1) = η n-1 (G n-1 )
This implies that

dη n-1 Q n dη n (x n ) × dη n-2 Q n-1 dη n-1 (x n-1 ) × • • • × dη 0 Q 1 dη 1 (x 1 ) = Z n Z n-1 × Z n-1 Z n-2 × • • • × Z 1 Z 0 = Z n RR n°7677
Using these observations, we readily prove the desired backward decomposition formula.

Semigroup models

This section is concerned with the semigroup structure and the weak regularity properties of Feynman-Kac models. Notice that Q p,n has the following functional representation

Q p,n (f n )(x p ) := E   f n (X n ) p≤q<n G q (X q ) | X p = x p   Definition 3.2
We let G p,n and P p,n be the potential functions and the Markov transitions defined by

Q p,n (1)(x) = G p,n (x) and P p,n (f ) = Q p,n (f ) Q p,n (1)
we also set g p,n := sup

x,y G p,n (x) G p,n (y) and β(P p,n ) = sup osc(P p,n (f ))

The r.h.s. supremum is taken the set of functions Osc(E). To simplify notation, for n = p + 1 we have also set

G p,p+1 = Q p,p+1 (1)
= G p and sometimes we write g p instead of g p,p+1 .

The particle concentration inequalities developed in chapter 6 will be expressed in terms of the following parameters. Definition 3.3 For any k, l ≥ 0, we also set

τ k,l (n) := 0≤p≤n g k p,n β(P p,n ) l and κ(n) := sup 0≤p≤n (g p,n β(P p,n )) (50)
Using the fact that

η n (f n ) := η p Q p,n (f n )/η p Q p,n (1) (51)
we prove the following formula

Φ p,n (η p ) = Ψ Gp,n (η p ) P p,n
for any 0 ≤ p ≤ n.

As a direct consequence of ( 19) and ( 20), we quote the following weak regularity property of the Feynman-Kac semigroups.

Φ p,n (µ) -Φ p,n (ν) = 1 ν(G p,n ) (µ -ν)S Gp,n,µ P p,n
In addition, we have the following Lipschitz estimates

Φ p,n (µ) -Φ p,n (ν) tv ≤ G p,n µ(G p,n ) ∨ ν(G p,n ) β(P p,n ) µ -ν tv and sup µ,ν Φ p,n (µ) -Φ p,n (ν) tv = β(P p,n )
3.4 Stability properties

Regularity conditions

In this section we present one of the simplest quantitative contraction estimate we known for the normalized Feynman-Kac semigroups Φ p,n . We consider the following regularity conditions.

H m (G, M) There exists some integer m ≥ 1, such that for any n ≥ 0, and any

((x, x ′ ), A) ∈ E 2 n × E n and any n ≥ 0 we have M n,n+m (x, A) ≤ χ m M n,n+m (x ′ , A) and g = sup n≥0 g n < ∞
for some finite parameters χ m , g < ∞, and some integer m ≥ 1.

H 0 (G, M) ρ := sup n≥0 (g n β(M n+1 )) < 1 and g = sup n g n < ∞ (52)
Both conditions are related to the stability properties of the reference Markov chain model X n with probability transition M n . They implies that the chain X n tends to merges exponentially fast the random states starting from any two different locations.

One natural strategy to obtain some useful quantitative contraction estimate for the Markov transitions P p,n is to write this transition in terms of the composition of Markov transitions. Lemma 3.1 For any 0 ≤ p ≤ q ≤ n, we have

P p,n = R (n)
p,q P q,n and P p,n = R

(n) p+1 R (n) p+2 . . . R (n) n-1 R (n) n
with the triangular array of Markov transitions R (n)

p,q and (R

(n) q ) 1≤q≤n defined by R (n) p,q (f ) := Q p,q (G q,n f ) Q p,q (G q,n ) = P p,q (G q,n f ) P p,q (G q,n )
and

R (n) p (f ) = Q p (G p,n f ) Q p (G p,n ) = M p (G p,n f ) M p (G p,n ) RR n°7677
In addition, for any 0 ≤ p ≤ q ≤ n we have β R (n) p,q ≤ g q,n β (P p,q ) and log g p,n ≤ p≤q<n (g q -1) β(P p,q )

Proof:

Using the decomposition

Q p,n (f ) = Q p,q (Q q,n (f )) = Q p,q (Q q,n (1) P q,n (f ))
we easily check the first assertion. The l.h.s. inequality in ( 53) is a direct consequence of [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF]. Using [START_REF] Carmona | Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses[END_REF], the proof of the r.h.s. inequality in ( 53) is based on the fact that

G p,n (x) G p,n (y) = p≤q<n Φ p,q (δ x )(G q ) p≤q<n Φ p,q (δ y )(G q ) = exp    p≤q<n (log Φ p,q (δ x )(G q ) -log Φ p,q (δ y )(G q ))    Using the fact that log y -log x = 1 0 (y -x) x + t(y -x) dt
for any positive numbers x, y, we prove that

G p,n (x) G p,n (y) = exp p≤q<n 1 0 (Φ p,q (δ x )(G q ) -Φ p,q (δ y )(G q )) Φ p,q (δ y )(G q ) + t(Φ p,q (δ x )(G q ) -Φ p,q (δ y )(G q )) dt ≤ exp    p≤q<n g q × Φ p,q (δ x )( G q ) -Φ p,q (δ y )( G q )    with G q := G q /osc(G q ) and g q := osc(G q )/inf G q ≤ g q -1
We end the proof of the desired estimates using [START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF], and proposition 3.1. This completes the proof of the lemma.

Quantitative contraction theorems

This section is mainly concerned with the proof of two contraction theorems that can be derived under the couple of regularity conditions presented in section 3.4.1.

RR n°7677

Theorem 3.1 We assume that condition H m (G, M) is satisfied for some finite parameters χ m , g < ∞, and some integer m ≥ 1. In this situation, we have the uniform estimates

sup 0≤p≤n g p,n ≤ χ m g m and sup p≥0 β(P p,p+km ) ≤ 1 -g -(m-1) χ -2 m k (54)
In addition, for any couple of measures ν, µ ∈ P(E p ), and for any f ∈ Osc(E n ) we have the decomposition

|[Φ p,n (µ) -Φ p,n (ν)](f )| ≤ ρ m (1 -κ m ) (n-p)/m |(µ -ν)D p,n,µ (f )| (55)
for some function D p,n,µ (f ) ∈ Osc(E p ) whose values only depends on the parameters (p, n, µ), and some parameters ρ m < ∞ and κ m ∈]0, 1] such that

ρ m ≤ χ m g m 1 -g -(m-1) χ -2 m -1 and κ m ≥ g -(m-1) χ -2 m ( 56
)
Proof:

For any non negative function f , we notice that

R (n) p,p+m (f )(x) = Q p,p+m (G p+m,n f )(x) Q p,p+m (G p+m,n )(x) ≥ g -(m-1) χ -2 m M p,p+m (G p+m,n f )(x ′ ) M p,p+m (G p+m,n )(x ′ )
and for any p

+ m ≤ n G p,n (x) G p,n (x ′ ) = Q p,p+m (G p+m,n )(x) Q p,p+m (G p+m,n )(x ′ ) ≤ g m M p,p+m (G p+m,n )(x) M p,p+m (G p+m,n )(x ′ ) ≤ χ m g m
For p ≤ n ≤ p + m, this upper bound remains valid. We conclude that

G p,n (x) ≤ χ m g m G p,n (x ′ ) and β R (n) p,p+m ≤ 1 -g -(m-1) χ -2 m
In the same way as above, we have

n = km ⇒ P p,p+km = R (n) p,p+m R (n) p+m,p+2m . . . R (n) p+(k-1)m,p+km and
β(P p,p+km ) ≤ 1≤l≤k β(R (n) p+(l-1)m,p+lm ) ≤ 1 -g -(m-1) χ -2 m k
This ends the proof of ( 54). The proof of ( 55) is based on the decomposition

(µ -ν)S Gp,n,µ P p,n (f ) = β(S Gp,n,µ P p,n ) × (µ -ν)D p,n,µ (f ) with D p,n,µ (f ) := S Gp,n,µ P p,n (f )/β(S Gp,n,µ P p,n )
On the other hand, we have

S Gp,n,µ (x, y) ≥ (1 -G p,n ) ⇒ β(S Gp,n,µ ) ≤ G p,n RR n°7677 and β(S Gp,n,µ )/ν(G p,n ) ≤ g p,n ≤ χ m g m
Finally, we observe that β(P p,n ) ≤ β(P p,p+⌊(n-p)/m⌋ ) from which we conclude that

1 ν(G p,n ) β(S Gp,n,µ P p,n ) ≤ χ m g m β(P p,p+⌊(n-p)/m⌋ )
The end of the proof is now a direct consequence of the contraction estimate [START_REF] Feller | Diffusion processes in genetics[END_REF]. This ends the proof of the theorem.

Theorem 3.2 We assume that condition H 0 (G, M) is satisfied for some ρ < 1. In this situation, for any couple of measures ν, µ ∈ P(E p ), and for any f ∈ Osc(E n ) we have the decomposition

|[Φ p,n (µ) -Φ p,n (ν)](f )| ≤ ρ (n-p) |(µ -ν)D p,n,µ (f )|
for some function D p,n,µ (f ) ∈ Osc(E p ), whose values only depends on the parameters (p, n, µ). In addition, for any 0 ≤ p ≤ n, we have the estimates β(P p,n ) ≤ ρ n-p and g p,n ≤ exp (g -1) (1ρ n-p )/(1ρ)

Proof: Using proposition 3.1, and recalling that β(S Gn-1,µ ) ≤ G n-1 , we readily prove that

|[Φ n (µ) -Φ n (ν)] (f )| ≤ g n-1 β(M n ) |(µ -ν)D n,µ (f )| ≤ ρ |(µ -ν)D n,µ (f )|
with the function

D n,µ (f ) = S Gn-1,µ M n (f )/β(S Gn-1,µ M n ) ∈ Osc(E n-1 )
Now, we can prove the theorem by induction on the parameter n ≥ p. For n = p, the desired result follows from the above discussion. Suppose we have

|[Φ p,n-1 (µ) -Φ p,n-1 (ν)](f )| ≤ ρ (n-p-1) |(µ -ν)D p,n-1,µ (f )|
for any f ∈ Osc(E n-1 ), and some functions D p,n-1,µ (f ) ∈ Osc(E p ). In this case, we have

|[Φ n (Φ p,n-1 (µ)) -Φ n (Φ p,n-1 (ν))](f )| ≤ g n-1 β(M n ) (Φ p,n-1 (µ) -Φ p,n-1 (ν))D n,Φp,n-1(µ) (f ) for any f ∈ Osc(E n ), with D n,Φp,n-1(µ) (f ) ∈ Osc(E n-1 ).
Under our assumptions, we conclude that

|[Φ n (Φ p,n-1 (µ)) -Φ n (Φ p,n-1 (ν))](f )| ≤ ρ (n-p) |(µ -ν)D p,n,µ (f )| RR n°7677
with the function

D p,n,µ (f ) := D p,n-1,µ D n,Φp,n-1(µ) (f ) ∈ Osc(E p )
The proof of the second assertion is a direct consequence of proposition 3.1, and lemma 3.1. This ends the proof of the theorem.

Corollary 3.1 Under any of the conditions H m (G, M), with m ≥ 0, the functions τ k,l and κ defined in (50) are uniformly bounded; that is, for any k, l ≥ 1 we have that

τ k,l (m) := sup n≥0 sup 0≤p≤n τ k,l (n) < ∞ and κ(m) := sup n≥0 κ(n) < ∞
In addition, for any m ≥ 1, we have

κ(m) ∈ [1, χ m g m ] τ k,l (m) ≤ m (χ m g m ) k / 1 -1 -g -(m-1) χ -2 m l
and for m = 0, we have the estimates

κ(0) ≤ exp ((g -1)/(1 -ρ)) τ k,l (0) ≤ exp (k(g -1)/(1 -ρ))/(1 -ρ l )

Some illustrations

We illustrate the regularity conditions presented in section 3.4.1 with three different types of Feynman-Kac models, related respectively to time discretization techniques, simulated annealing type schemes, and path space models. Of course, a complete analysis of the regularity properties of the 20 Feynman-Kac application models presented in section 2 would lead to a too long discussion.

In some instances, the regularity conditions stated in section 3.4.1 can be directly translated into regularity properties of the reference Markov chain model and the adaptation potential function.

In other instances, the regularity properties of the Feynman-Kac semigroup depend on some important tuning parameters, including discretization time steps, and cooling schedule in simulated annealing time models. In section 3.4.4, we illustrate the regularity property H 0 (G, M) stated in [START_REF] Everett | Los Alamos report[END_REF] in the context of time discretization model with geometric style clocks introduced in (42). In section 3.4.5, we present some tools to tune the cooling parameters of the annealing model discussed in [START_REF] Del Moral | Sequential Monte Carlo samplers[END_REF], so that the resulting semigroups are exponentially stable.

For degenerate indicator style functions, we can use a one step integration technique to transform the model into a Feynman-Kac model on smaller state spaces with positive potential functions. In terms of particle absorption models, this technique allows to turn a hard obstacle model into a soft obstacle particle model. Further details on this integration technique can be found in [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Particle Motions in Absorbing Medium with Hard and Soft Obstacles[END_REF].

Last, but not least, in some important applications, including Feynman-Kac models on path spaces, the limiting semigroups are unstable, in the sense that they don't forget their initial conditions. Nevertheless, in some situations it is still possible to control uniformly in time the quantities g p,n discussed in section 3.3

Time discretization models

We consider the potential functions G n and Markov transitions M n are given by [START_REF] Del Moral | On a class of discrete generation interacting particle systems[END_REF], for some non negative function V n , some positive parameter λ n and some Markov transition K n s.t.

β(K n ) ≤ κ n < 1 h ≤ h n = (1 -κ n )/[v n-1 + α] and λ n ∈]0, 1/h] with v n-1 := osc(V n-1
), and for some α > 0. We also assume that v = sup n v n < ∞.

In this situation, for any λ n ∈ 1 hn , 1 h , we have

g n-1 β(M n ) ≤ e vn-1h (1 -λ n h (1 -κ n )) ≤ e -h(λn(1-κn)-vn-1) ≤ e -αh
from which we conclude that H 0 (G, M) is met with g = sup n g n ≤ e hv and ρ ≤ e -αh

Interacting simulated annealing model

We consider the Feynman-Kac annealing model discussed in [START_REF] Del Moral | Sequential Monte Carlo samplers[END_REF]. We further assume that K kn n (x, dy) ≥ ǫ n ν n (y) for some k n ≥ 1, some ǫ n > 0, and some measure ν n .

In this situation, we have M kn n,βn (x, dy) ≥ K kn n (x, dy) e -βnknv ≥ ǫ n e -βnknv ν n (dy)

with v := osc(V ). if we choose m n = k n l n , this implies that β(M n ) = β M mn n,βn ≤ β M kn n,βn ln ≤ 1 -ǫ n e -βnknv ln
Therefore, for any given ρ ′ ∈]0, 1[ we can chose l n such that

l n ≥ log (1/ρ ′ ) + v(β n -β n-1 ) log 1/(1 -ǫ n e -βnknv ) so that g n-1 β(M n ) ≤ e v(βn-βn-1) × 1 -ǫ n e -βnknv ln ≤ ρ ′ ⇒ ρ ≤ ρ ′ For any function β : x ∈ [0, ∞[ → β(x)
, with a decreasing derivative β ′ (x) s.t. lim x→∞ β ′ (x) = 0 and β ′ (0) < ∞, we also notice that

g = sup n≥0 g n ≤ sup n≥0 e vβ ′ n ≤ e vβ ′ (0)
RR n°7677

Historical processes

We return to the historical Feynman-Kac models introduced in section 3.2.

Using the equivalence principle (47), we have proved that the n-time marginal models associated with a Feynman-Kac model on path space coincide with the original Feynman-Kac measure (1). We write Q p,n and P p,n the Feynman-Kac semigrousp defined as Q p,n and P p,n , by replacing (G n , M n ) by (G n , M n ). By construction, we have

Q p,n (1)(x p ) = Q p,n (1)(x p )
for any x p = (x 0 , . . . , x p ) ∈ E p . Therefore, if we set

G p,n (x p ) := Q p,n (1)(x p )
then we find that

g p,n := sup xp,yp G p,n (x p ) G p,n (y p ) = sup xp,yp G p,n (x p ) G p,n (y p ) = g p,n
On the other hand, we cannot expect the Dobrushin's ergodic coefficient of the historical process semigroup to decrease but we always have have β(P p,n ) ≤ 1.

In summary, when the reference Markov chain X n satisfies the condition H m (G, M) stated in the beginning of section 3.4.1, for some m ≥ 1, we always have the estimates g p,n ≤ χ m g m and β(P p,n ) ≤ 1 [START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF] and

τ k,l (n) ≤ (n + 1)(χ m g m ) k and κ(n) ≤ χ m g m (58)
with the functions τ k,l , and κ introduced in (50)

We end this section with some Markov chain Monte Carlo technique often used in practice to stabilize the genealogical tree based approximation model. To describe with some precision this stochastic method, we consider the Feynman-Kac measures η n ∈ P(E n ) associated with the potential function G n and the Markov transitions M n of the historical process defined respectively in [START_REF] Di Masi | An approximation for the nonlinear filtering problem with error bounds[END_REF] and in [START_REF] Moral | A note on Laplace-Varadhan's integral lemma[END_REF]. We notice that η n satisfy the updating-prediction equation

η n = Ψ Gn (η n )M n
This equation on the set of measures on path spaces is unstable, in the sense that its initial condition is always kept in memory by the historical Markov transitions M n . One idea to stabilize this system is to incorporate an additional Markov chain Monte Carlo move at every time step. More formally, let us suppose that we have a dedicated Markov chain Monte Carlo transition K n from the set E n into itself, and such that

η n = η n K n
In this situation, we also have that

η n = Ψ Gn (η n )M ′ n with M ′ n := M n K n (59)

RR n°7677

By construction, the mean field particle approximation of the equation ( 59) is a genealogical tree type evolution model with path space particles on the state spaces E n . The updating-selection transitions are related to the potential function G n on the state spaces E n , and the mutation-exploration mechanisms from E n into E n+1 are dictated by the Markov transitions M ′ n+1 . Notice that this mutation transition is decomposed into two different stages. Firstly, we extend the selected path-valued particles with an elementary move according to the Markov transition M n . Then, from every of these extended paths, we perform a Markov chain Monte Carlo sample according to the Markov transition K n .

Mean field particle models

Interacting particle systems

With the exception of some very special cases, the measures η n cannot be represented in a closed form, even on in finite dimensional state-spaces. Their numerical estimation using deterministic type grid approximations requires extensive calculations, and their rarely cope with high dimensional problems. In the same vein, harmonic type approximation schemes, or related linearization style techniques such as the extended Kalman filter often provide poor estimations result for highly nonlinear models. In contrast with these conventional techniques, mean field particle models can be thought as a stochastic adaptive grid approximation scheme. These advanced Monte Carlo methods take advantage of the nonlinearities of the model, so that to design an interacting selection-recycling mechanism.

Formally speaking, discrete generation mean field particle models are based on the fact that the flow of probability measures η n satisfy a non linear evolution equation of the following form

η n+1 (dy) = η n (dx)K n+1,ηn (x, dy) (60)
for some collection of Markov transitions K n+1,η , indexed by the time parameter n ≥ 0 and the set of probability measures P(E n ).

The choice of the McKean transitions K n+1,ηn is not unique. For instance, we can choose

K n+1,ηn (x, dy) = Φ n+1 (η n )(dy)
and more generally

K n+1,ηn (x, dy) = ǫ(η n )G n (x) M n+1 (x, dy) + (1 -ǫ(η n )G n (x)) Φ n+1 (η n )(dy) for any ǫ(η n ) s.t. ǫ(η n )G n (x) ∈ [0, 1]
. Note that we can define sequentially a Markov chain sequence (X n ) n≥0 such that

P X n+1 ∈ dx | X n = K n+1,ηn X n , dx with Law(X n ) = η n
From the practical point of view, this Markov chain can be seen as a perfect sampler of the flow of the distributions (60) of the random states X n . For a

RR n°7677

more thorough discussion on these nonlinear Markov chain models, we refer the reader to section 2.5 in the book [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF].

The mean field particle interpretation of this nonlinear measure valued model is the E N n -valued Markov chain

ξ n = ξ 1 n , ξ 2 n , . . . , ξ N n ∈ E N n
with elementary transitions defined as

P (ξ n+1 ∈ dx | ξ n ) = N i=1 K n+1,η N n (ξ i n , dx i ) (61)
with

η N n := 1 N N j=1 δ ξ j n
In the above displayed formula, dx stands for an infinitesimal neighborhood of the point x = (x 1 , . . . , x N ) ∈ E N n+1 . The initial system ξ 0 consists of N independent and identically distributed random variables with common law η 0 .

We let G N n := σ (ξ 0 , . . . , ξ n ) be the natural filtration associated with the N -particle approximation model defined above.

The particle model associated with the parameter ǫ(η n ) = 1 coincides with the genetic type stochastic algorithm presented in section 1.4.

Furthermore, using the equivalence principles (47) presented in section 3.2, we can check that the genealogical tree model discussed above coincides with the mean field N -particle interpretation of the Feynman-Kac measures (γ n , η n ) associated with the pair (G n , M n ) on the path spaces E n . In this context, we recall that η n = Q n , and the N -particle approximation measures are given by

η N n := 1 N N i=1 δ (ξ i 0,n ,ξ i 1,n ,...,ξ i n,n ) ∈ P(E n ) = P(E 0 × . . . × E n ) (62)

Local sampling errors

The local sampling errors induced by the mean field particle model [START_REF] Holland | Adaptation in Natural and Artificial Systems[END_REF] are expressed in terms of the empirical random field sequence V N n defined by

V N n+1 = √ N η N n+1 -Φ n+1 η N n
Notice that V N n+1 is alternatively defined by the following stochastic perturbation formulae

η N n+1 = Φ n+1 η N n + 1 √ N V N n+1 (63)
For n = 0, we also set

V N 0 = √ N η N 0 -η 0 ⇔ η N 0 = η 0 + 1 √ N V N 0
In this interpretation, the N -particle model can also be interpreted as a stochastic perturbation of the limiting system

η n+1 = Φ n+1 (η n ) RR n°7677
It is rather elementary to check that

E V N n+1 (f ) G N n = 0 E V N n+1 (f ) 2 G N n = η N n K n+1,η N n f -K n+1,η N n (f ) 2 Definition 3.4
We denote by σ 2 n the uniform local variance parameter given by

σ 2 n := sup µ K n,µ [f n -K n,µ (f n )] 2 ≤ 1 (64)
In the above displayed formula the supremum is taken over all functions f n ∈ Osc(E n ), and all probability measures µ on E n , with n ≥ 1. For n = 0, we set

σ 2 0 = sup f0∈Osc(E0) η 0 [f 0 -η 0 (f 0 )] 2 ≤ 1
We close this section with a brief discussion on these uniform local variance parameters in the context of continuous time discretization models. When the discrete time model K n,µ = K (h) n,µ comes from a discretization of the continuous time model with time step ∆t = h(≤ 1), we often have that

K n,µ = Id + hL n,µ + O(h 2 ) (65)
for some infinitesimal generator L n,µ . In this situation, we also have that

L Kn,µ := K n,µ -Id = hL n,µ + O(h 2 )
For any Markov transition K, we notice that

K([f -K(f )] 2 ) = K(f 2 ) -K(f ) 2 = L K (f 2 ) -2f L K (f ) -(L K (f )) 2 = Γ LK (f, f ) -(L K (f )) 2
with the carré du champ" function Γ LK (f, f ) defined for any x ∈ E by

Γ LK (f, f )(x) = L K [f -L K (f )(x)] 2 (x) = L K (f 2 )(x) -2f (x) L K (f )(x) When K = K n,µ and L Kn,µ = hL n,µ + O(h 2 ), we find that µ K n,µ [f n -K n,µ (f n )] 2 = µΓ LK n,µ (f, f ) h -h 2 µ(L Kn,µ (f ) 2 ) = h µ Γ Ln,µ (f, f ) + O(h 2 )
4 Empirical processes

Introduction

The aim of this chapter is to review some more or less well known stochastic techniques for analyzing the concentration properties of empirical processes associated with independent random sequences. The discussion at the start of this section provides some basic definitions on empirical processes associated with sequences of independent random variables on general measurable state spaces. In section 4.2, we state and comment the main results of this section. Section 4.2.1 is concerned with finite marginal models. In section 4.2.2, we extend these results at the level of the empirical processes. Besides the fact that the concentration inequalities for empirical processes holds for supremum of empirical processes over infinite collection of functions, these inequalities are more crude with greater constants than the ones for marginal models. These two sections also contains two new perturbation theorems that apply to nonlinear functional of empirical processes. The proofs of these theorems combine Orlicz's norm techniques, Kintchine's type inequalities, maximal inequalities, as well as Laplace-Cramèr-Chernov estimation methods. These 4 complementary methodologies are presented respectively in section 4.3, section 4.4, section 4.5, and section 4.6. Let (µ i ) i≥1 be a sequence of probability measures on a given measurable state space (E, E). During the further development of this section, we fix an integer N ≥ 1. To clarify the presentation, we slightly abuse the notation and we denote respectively by

m(X) = 1 N N i=1 δ X i and µ = 1 N N i=1 µ i
the N -empirical measure associated with a collection of independent random variables X = (X i ) i≥1 , with respective distributions (µ i ) i≥1 , and the N -averaged measure associated with the sequence of measures (µ i ) i≥1 . We also consider the empirical random field sequences

V (X) = √ N (m(X) -µ)
We also set

σ(f ) 2 := E V (X)(f ) 2 = 1 N N i=1 µ i ([f -µ i (f )] 2 ) ( 66
)
Remark 4.1 The rather abstract models presented above can be used to analyze the local sampling random fields models associated with a mean field particle model discussed in section 3.6.

To be more precise, given the information on the N -particle model at time (n -1), the sequence of random variables ξ i n are independent random sequences with a distribution that depends on the current state ξ i n-1 . That is, at any given fixed time horizon n and given G N n-1 , we have

X i = ξ i n ∈ E = E n and µ i (dx) := K n,η N n-1 (ξ i n-1 , dx) (67)
In this case, we find that

m(X) = η N n and V (X) = V N n and σ(f ) 2 = E V N n+1 (f ) 2 G N n = η N n K n+1,η N n f -K n+1,η N n (f ) 2 RR n°7677
Let F be a given collection of measurable functions f : E → R such that f ≤ 1. We associate with F the Zolotarev seminorm on P(E) defined by

µ -ν F = sup |µ(f ) -ν(f )|; f ∈ F ,
(see for instance [START_REF] Rachev | Probability Metrics and the Stability of Stochastic Models[END_REF]). No generality is lost and much convenience is gained by supposing that the unit and the null functions f = 1 and f = 0 ∈ F . Furthermore, to avoid some unnecessary technical measurability questions, we shall also suppose that F is separable in the sense that it contains a countable and dense subset. We measure the size of a given class F in terms of the covering numbers N (ε, F , L 2 (µ)) defined as the minimal number of L 2 (µ)-balls of radius ε > 0 needed to cover F . We shall also use the following uniform covering numbers and entropies.

We end this section with the last of the notation to be used in this chapter dedicated to empirical processes concentration inequalities. Definition 4.1 By N (ε, F ), ε > 0, and by I(F ) we denote the uniform covering numbers and entropy integral given by

N (ε, F ) = sup N (ε, F , L 2 (η)); η ∈ P(E) I(F ) = 2 0 log (1 + N (ε, F )) dε
The concentration inequalities stated in this section are expressed in terms of the inverse of the couple of functions defined below. Definition 4.2 We let (ǫ 0 , ǫ 1 ) be the functions on R + defined by

ǫ 0 (λ) = 1 2 (λ -log (1 + λ)) and ǫ 1 (λ) = (1 + λ) log (1 + λ) -λ
Rather crude estimates can be derived using the following upper bounds

ǫ -1 0 (x) ≤ 2(x + √ x) and ǫ -1 1 (x) ≤ x 3 + √ 2x
A proof of these elementary inequalities and refined estimates can be found in the recent article [START_REF] Moral | Concentration inequalities for mean field particle models[END_REF].

Statement of the main results

Finite marginal models

The main result of this section is a quantitative concentration inequality for the finite marginal models

f → V (X)(f ) = √ N (m(X) -µ(f ))
In the following theorem, we provide Kintchine's type mean error bounds, and related Orlicz norm estimates. The detailed proofs of these results are housed in section 4.4. The last quantitative concentration inequality is a direct consequence of ( 70), and it is proved in remark 4.6.

E(|V (X)(f )| m ) 1/m ≤ b(m) osc(f ) ∧ 2 µ(|f | m ′ ) 1/m ′ ( 68
)
and

E (|V (X)(f )| m ) 1 m ≤ 6b(m) 2 max √ 2σ(f ), 2σ(f ) 2 N m ′ 2 -1 1/m ′
with the smallest even integer m ′ ≥ m, and the collection of constants b(m) defined in [START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF].

In particular, for any f ∈ Osc(E), we have

π ψ (V (X)(f )) ≤ 3/8 ( 69
)
and for any N s.t. 2σ 2 (f )N ≥ 1 we have

E (|V (X)(f )| m ) 1 m ≤ 6 √ 2 b(m) 2 σ(f ) (70)
In addition, the probability of the event is greater than 1e -y , for any y ≥ 0 (cf. lemma 4.2). This implies that the probability of the events

|V (X)(f )| ≤ 6 √ 2 σ(f ) 1 + ǫ -1 0 (x) is greater than 1 -e -x ,
|V (X)(f )| ≤ 1 2 3(x + log 2)/2
is greater than 1e -x , for any x ≥ 0. Ou next objective is to derive concentration inequalities for nonlinear functional of the empirical random field V (X). To introduce precisely these objects, we need another around of notation.

For any measure ν, and any sequence of measurable functions f = (f 1 , . . . , f d ), we write ν(f

) := [ν(f 1 ), . . . , ν(f d )]
Definition 4.3 We associate with the second order smooth function F on R d , for some d ≥ 1, the random functionals defined by

f = (f i ) 1≤i≤d ∈ Osc(E) d → F (m(X)(f )) = F (m(X)(f 1 ), . . . , m(X)(f d )) ∈ R (71)
Given a probability measure ν, and a collection of functions

(f i ) 1≤i≤d ∈ Osc(E) d , we set D ν (F )(f ) = ∇F (ν(f )) f ⊤ (72) Notice that osc (D ν (F )(f )) ≤ ∇F (ν(f )) 1 := d i=1 ∂F ∂u i (ν(f ))
We also introduce the following constants

∇ 2 F f 1 := d i,j=1 sup ∂ 2 F ∂u i ∂u j (ν(f )) (73)
In the r.h.s. display, the supremum is taken over all probability measures ν ∈ P(E).

The next theorem extend the exponential inequalities stated in theorem 4.1 to this class of nonlinear functionals. It also provide more precise concentration properties in terms of the variance functional σ defined in [START_REF] Korezlioglu | Filtering for nonlinear systems driven by nonwhite noises : An approximating scheme[END_REF]. The proof of this theorem is postponed to section 4.7.

Theorem 4.2 Let F be a second order smooth function on R d , for some d ≥ 1. For any collection of functions (f i ) 1≤i≤d ∈ Osc(E) d , and any N ≥ 1, the probability of the events

[F (m(X)(f )) -F (µ(f ))] ≤ 1 2N ∇ 2 F f 1 3/2 + ǫ -1 0 (x) + ∇F (µ(f )) -1 1 σ 2 (D µ (F )(f )) ǫ -1 1 x ∇F (µ(f )) 2 1 N σ 2 (D µ (F )(f ))
is greater than 1e -x , for any x ≥ 0. In the above display, D µ (F )(f ) stands for the first order function defined in (72).

Empirical processes

The objective of this section is to extend the quantitative concentration theorems, theorem 4.1 and theorem 4.2 at the level of the empirical process associated with a class of function F . These processes are given by the mapping

f ∈ F → V (X)(f ) = √ N (m(X) -µ(f ))
Our main result in this direction is the following theorem, whose proof is postponed to section 4.5.

Theorem 4.3 For any class of functions F , with I(F ) < ∞, we have

π ψ ( V (X) F ) ≤ 12 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ
Remark 4.2 Using the fact that log (8 + x 2 ) ≤ 4 log x, for any x ≥ 2, we obtain the rather crude estimate

2 0 log (8 + N (F , ǫ) 2 ) dǫ ≤ 2 2 0 log N (F , ǫ) dǫ
We check the first observation, using the fact that θ(x) = 4 log xlog (8 + x 2 ) is a non decreasing function on R + , and θ(2

) = log (4 × 4) -log (4 × 3) ≥ 0.
Various examples of classes of functions with finite covering and entropy integral are given in the book of Van der Vaart and Wellner [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes with Applications to Statistics[END_REF] (see for instance p. 86, p. 129, p. 135, and exercise 4 on p.150, and p. 155). The estimation of the quantities introduced above often depends on several deep results on combinatorics that are not discussed here.

To illustrate these mathematical objects, we mention that, for the set of indicator functions

F = 1 d i=1 (-∞,xi] ; (x i ) 1≤i≤d ∈ R d ( 74
)
of cells in E = R d , we have

N (ε, F ) ≤ c (d + 1)(4e) d+1 ǫ -2d
for some universal constant c < ∞. This implies that

log N (ε, F ) ≤ log [c(d + 1)(4e) d+1 ] + (2d) log (1/ǫ)
An elementary calculation gives

2 0 log (1/ǫ) ≤ 2 ∞ 0 x 2 e -x 2 dx = π/4 ≤ 1 from which we conclude that 2 0 log N (F , ǫ) dǫ ≤ 2 log [c(d + 1)(4e) d+1 ] + (2d) ≤ c ′ √ d ( 75
)
for some universal constant c < ∞.

For d = 1, we also have that N (ε, F ) ≤ 2/ǫ 2 (cf. p. 129 in [START_REF] Van Der Vaart | Weak Convergence and Empirical Processes with Applications to Statistics[END_REF]) and therefore

2 0 log N (F , ǫ) dǫ ≤ 3 √ 2
Remark 4.3 In this chapter, we have assumed that the class of functions F is such that sup f ∈F f ≤ 1. When sup f ∈F f ≤ c F , for some finite constant c F , using theorem 4.3, it is also readily checked that

π ψ ( V (X) F ) ≤ 12 2 2cF 0 log (8 + N (F , ǫ) 2 ) dǫ (76)
We mention that the uniform entropy condition I(F ) < ∞ is required in Glivenko-Cantelli and Donsker theorems for empirical processes associated with non necessarily independent random sequences [START_REF] Del Moral | On the Convergence and the Applications of Empirical Processes for Interacting Particle Systems and Nonlinear Filtering[END_REF].

Arguing as above, we prove that the probability of the events

V (X) F ≤ I 1 (F ) x + log 2
is greater than 1e -x , for any x ≥ 0, with some constant

I 1 (F ) ≤ 12 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ RR n°7677
As for marginal models 71, our next objective is to extend theorem 4.5 to empirical processes associated with some classes of functions. Here, we consider the empirical processes

f ∈ F i → m(X)(f ) ∈ R
associated with d classes of functions F i , 1 ≤ i ≤ d, defined in section 4.1. We further assume that f i ∨ osc(f i ) ≤ 1, for any f i ∈ F i , and we set

F := 1≤i≤d F i and π ψ ( V (X) F ) := sup 1≤i≤d π ψ ( V (X) Fi )
Using theorem 4.3, we mention that

π ψ ( V (X) F ) ≤ 12 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ with N (F , ǫ) := sup 1≤i≤d N (F i , ǫ)
We set

∇F µ ∞ := sup ∂F ∂u i (µ(f )) and ∇ 2 F ∞ = sup ∂ 2 F ∂u i ∂u j (ν(f ))
The supremum in the l.h.s. is taken over all 1 ≤ i ≤ d and all f ∈ F ; and the supremum in the r.h.s. is taken over all 1 ≤ i, j ≤ d, ν ∈ P(E), and all f ∈ F.

We are now in position to state the final main result of this section. The proof of the next theorem is housed in the end of section 4.7.

Theorem 4.4 Let F be a second order smooth function on R d , for some d ≥ 1. For any classes of functions F i , 1 ≤ i ≤ d, and for any x ≥ 0, the probability of the following events

sup f ∈F |F (m(X)(f )) -F (µ(f ))| ≤ d √ N π ψ ( V (X) F ) ∇F µ ∞ 1 + 2 √ x + 1 2N ∇ 2 F ∞ (d π ψ ( V (X) F )) 2 1 + ǫ -1 0 x 2 is greater than 1 -e -x .

A reminder on Orlicz' norms

In this section, we have collected some important properties of Orlicz' norms. The first section, section 4.3.1 is concerned with rather elementary comparison properties. In section 4.3.2, we present a natural way to obtain Laplace estimates, and related concentration inequalities, using simple Orlicz' norm upper bounds.

Comparison properties

This short section is mainly concerned with the proof of the following three comparison properties.

Lemma 4.1 For any non negative variables (Y 1 , Y 2 ) we have

Y 1 ≤ Y 2 =⇒ π ψ (Y 1 ) ≤ π ψ (Y 2 )
as well as

∀m ≥ 0 E Y 2m 1 ≤ E Y 2m 2 ⇒ π ψ (Y 1 ) ≤ π ψ (Y 2 ) (77)
In addition, for any pair of independent random variables (X, Y ) on some measurable state space, and any measurable function f , we have

(π ψ (f (x, Y )) ≤ c for P-a.e. x) =⇒ π ψ (f (X, Y )) ≤ c ( 78
)
Proof:

The first assertion is immediate, ad the second assertion comes from the fact that

E exp Y 1 π ψ (Y 2 ) 2 -1 ≤ m≥1 1 m! E(Y 2m 2 ) π ψ (Y 2 ) 2m = E Y 2 π ψ (Y 2 ) ≤ 1
The last assertion comes from the fact that

E (E (ψ(f (X, Y )/c) |X )) ≤ 1 ⇒ π ψ (f (X, Y )) ≤ c
This ends the proof of the lemma.

Concentration properties

The following lemma provides a simple way to transfer a control on Orlicz' norm into moment or Laplace estimates, which in turn can be used to derive quantitative concentration inequalities Lemma 4.2 For any non negative random variable Y , and any integer m ≥ 0, we have

E Y 2m ≤ m! π ψ (Y ) 2m and E Y 2m+1 ≤ (m + 1)! π ψ (Y ) 2m+1 (79)
In addition, for any t ≥ 0 we have the Laplace estimates

E e tY ≤ min 2 e 1 4 (tπ ψ (Y )) 2 , (1 + tπ ψ (Y )) e (tπ ψ (Y )) 2
In particular, for any x ≥ 0 the probability of the event

Y ≤ π ψ (Y ) x + log 2 (80)
is greater than 1e -x .

RR n°7677

Remark 4.4 For a Gaussian and centred random variable Y , s.t. E(Y 2 ) = 1, we recall that π ψ (Y ) = 8/3. In this situation, letting y = 8(x + log 2)/3 in (80), we find that

P (|Y | ≥ y) ≤ 2 e -1 2 3 4 y 2
Working directly with the Laplace Gaussian function E e tY = e t 2 /2 , we remove the factor 3/4. In this sense, we loose a factor 3/4 using the Orlicz's concentration property [START_REF] Pulvirenti | Kinetic limits for stochastic particle systems[END_REF].

In this situation, the l.h.s. moment estimate in ( 79) takes the form

b(2m) 2m = (2m)! m! 2 -m ≤ m! (8/3) m ( 81
)
while using Stirling's approximation of the factorials we obtain the estimate

(2m)! m! 2 ≃ 2/m 4 m (≤ (8/3) m )
Remark 4.5 Given a sequence of independent Gaussian and centred random variables Y i , s.t. E(Y 2 i ) = 1, for i ≥ 1, and any sequence of non negative numbers a i , we have

π ψ n i=1 a i Y i = 8/3 n i=1 a 2 i := 8/3 a 2 while n i=1 a i π ψ (Y i ) = 8/3 n i=1 a i := 8/3 a 1 Notice that a 2 ≤ a 1 ≤ √ n a 2
When the coefficients a i are almost equal, we can loose a factor √ n using the triangle inequality, instead of estimating directly with the Orlicz norm of the Gaussian mixture. In this sense, it is always preferable to avoid the use of the triangle inequality, and to estimate directly the Orlicz norms of linear combinations of "almost Gaussian" random variables.

Now, we come to the proof of the lemma.

Proof of lemma 4.2:

For any m ≥ 1, we have

x 2m ≤ m! n≥1 x 2n n! = m! ψ(x) ⇓ E Y π ψ (Y ) 2m ≤ m! E ψ Y π ψ (Y ) ≤ m!
For odd integers, we simply use Cauchy-Schwartz inequality to check that

E Y 2m+1 2 ≤ E Y 2m E Y 2(m+1) ≤ (m + 1)! 2 π ψ (Y ) 2(2m+1)

RR n°7677

This ends the proof of the first assertion. We use Cauchy-Schwartz's inequality to check that

E Y 2m+1 2 ≤ E Y 2m E Y 2(m+1) ≤ (m + 1)! 2 π ψ (Y ) 2(2m+1)
for any non negative random variable Y , so that

E Y 2m+1 ≤ (m + 1)! π ψ (Y ) (2m+1)
Recalling that (2m)! ≥ m! 2 and (m + 1) ≤ (2m + 1), we find that

E e tY = m≥0 t 2m (2m)! E Y 2m + m≥0 t 2m+1 (2m + 1)! E Y 2m+1 ≤ m≥0 t 2m m! π ψ (Y ) 2m + m≥0 t 2m+1 m! π ψ (Y ) (2m+1) = (1 + tπ ψ (Y )) exp (tπ ψ (Y )) 2
On the other hand, using the estimate

tY = tπ ψ (Y ) √ 2 √ 2 Y π ψ (Y ) ≤ (tπ ψ (Y )) 2 4 + Y π ψ (Y ) 2 we prove that E e tY ≤ 2 exp (tπ ψ (Y )) 2 4
The end of the proof of the Laplace estimates is now completed. To prove the last assertion, we use the fact that for any y ≥ 0

P (Y ≥ y) ≤ 2 exp -sup t≥0 ty -(tπ ψ (Y )) 2 /4 = 2 exp -(y/π ψ (Y )) 2 This implies that P Y ≥ π ψ (Y ) x + log 2 ≤ 2 exp [-(x + log 2)] = e -x
This ends the proof of the lemma.

Maximal inequalities

Let us now put together the Orlicz's norm properties derived in section 4.3 to establish a series of more or less well known maximal inequalities. More general results can be found in the books [START_REF] Pollard | Empirical processes : theory and applications[END_REF][START_REF] Van Der Vaart | Weak Convergence and Empirical Processes with Applications to Statistics[END_REF], or in the lecture notes [START_REF] Giné | Lectures on some aspects of the bootstrap[END_REF].

We emphasize that in the literature on empirical processes, maximal inequalities are often presented in terms of universal constant c without further information on their magnitude. In the present section, we shall try to estimate some of these universal constants explicitly.

To begin with, we consider a couple of maximal inequalities over finite sets.

Lemma 4.3 For any finite collection of non negative random variables (Y i ) i∈I , and any collection of non negative numbers (a i ) i∈I , we have

sup i∈I E(ψ(Y i /a i )) ≤ 1 ⇒ E max i∈I Y i ≤ ψ -1 (|I|) × max i∈I a i
Proof:

We check this claim using the following estimates

ψ E (max i∈I Y i ) max i∈I a i ≤ ψ E max i∈I (Y i /a i ) ≤ E ψ max i∈I (Y i /a i ) ≤ E i∈I ψ ((Y i /a i )) ≤ |I|
This ends the proof of the lemma.

Working a little harder, we prove the following lemma.

Lemma 4.4 For any finite collection of non negative random variables (Y i ) i∈I , we have

π ψ max i∈I Y i ≤ 6 log (8 + |I|) max i∈I π ψ (Y i ) Proof:
Without lost of generality, we assume that max i∈I π ψ (Y i ) ≤ 1, and I = {1, . . . , |I|}.

In this situation, it suffices to check that

ψ max 1≤i≤|I| Y i 6 log (8 + |I|) ≤ ψ max 1≤i≤|I| Y i 6 log (8 + i) ≤ 1
Firstly, we notice that for any i ≥ 1 and x ≥ 3/2 we have

1 log (8 + i) + 1 log x ≤ 1 log 9 + 1 log (3/2) ≤ 3
and therefore

3 log (8 + i) log (x) ≥ log (x(8 + i))
We check the first estimate using the fact that

log(3) ≤ 5 log(3/2) ⇒ log(3) + log(3/2) ≤ 6 log(3/2) ≤ 3 log(3/2) log(9)
Using these observations, we have

P max 1≤i≤|I| Yi √ 6 log (8+i) 2 > log x = P max 1≤i≤|I| Yi √ 6 log (x) log (8+i) 2 > 1 ≤ P max 1≤i≤|I| Yi √ 2 log (x(8+i)) > 1 ≤ |I| i=1 P Y i > 2 log (x(8 + i)) ≤ |I| i=1 e -2 log (x(8+i)) E e Y 2 i

RR n°7677

This implies that

P max 1≤i≤|I| Yi √ 6 log (8+i) 2 > log x ≤ 2 x 2 |I| i=1 1 (8 + i) 2 ≤ 2 x 2 ∞ 8 1 u 2 du = 1 (2x) 2
If we set

Z I := exp    max 1≤i≤|I| Y i 6 log (8 + i) 2    then we have E (Z I ) = ∞ 0 P (Z I > x) dx ≤ 3 2 + ∞ 3 2 1 (2x) 2 dx = 3 2 1 + 1 4 = 15 8 ≤ 2
and therefore

ψ max 1≤i≤|I| Y i 6 log (8 + i) ≤ 1
This ends the proof of the lemma.

The following technical lemma is pivotal in the analysis of maximal inequalities for sequences of random variables indexed by infinite but separable subsets equipped with a pseudo-metric, under some Lipschitz regularity conditions w.r.t. the Orlicz's norm. Lemma 4.5 We assume that the index set (I, d) is a separable, and totally bounded pseudo-metric space, with finite diameter

d(I) := sup (i,i)∈I 2 d(i, j) < ∞
We let (Y i ) i∈I be a separable and R-valued stochastic process indexed by I and such that

π ψ (Y i -Y j ) ≤ c d(i, j)
for some finite constant c < ∞. We also assume that Y i0 = 0, for some i 0 ∈ I. Then, we have

π ψ sup i∈I Y i ≤ 12 c d(I) 0 6 log (8 + N (I, d, ǫ) 2 ) dǫ Proof:
Replacing Y i by Y i /d(I), and d by d/d(I), there is no loss of generality to assume that d(I) ≤ 1. In the same way, Replacing Y i by Y i /c, we can also assume that c ≤ 1. For a given finite subset J ⊂ I, with i 0 ∈ J, we let J k = {i k 1 , . . . , i k n k } ⊂ J, be the centers of n k = N (J, d, 2 -k ) balls of radius at most 2 -k covering J. For k = 0, we set J 0 = {i 0 }. We also consider the mapping

θ k : i ∈ J → θ k (i) ∈ J k s.t. sup i∈J d(θ k (i), i) ≤ 2 -k
The set J being finite, there exist some sufficiently integer k ⋆ J s.t. d(θ k (i), i) = 0, for any k ≥ k ⋆ J ; and therefore Y i = Y θ k (i) , for any i ∈ J, and any k ≥ k ⋆ J . This implies that

Y i = k ⋆ J k=1 Y θ k (i) -Y θ k-1 (i)
We also notice that

d(θ k (i), θ k-1 (i)) ≤ d(θ k (i), i) + d(i, θ k-1 (i)) ≤ 2 -k + 2 -(k-1) = 3 × 2 -k
and sup

(i,j)∈(J k ×J k-1 ) : d(i,j)≤3×2 -k π ψ (Y i -Y j ) ≤ 3 × 2 -k
Using lemma 4.4 we prove that

π ψ sup i∈J Y i ≤ k ⋆ J k=1 π ψ sup i∈J Y θ k (i) -Y θ k-1 (i) ≤ 3 k ⋆ J k=1 6 log (8 + N (J, d, 2 -k ) 2 ) 2 -k
On the other hand, we have

2 2 -k -2 -(k+1) = 2 -k and 6 log (8 + N (J, d, 2 -k ) 2 ) 2 -k ≤ 2 2 -k 2 -(k+1)
6 log (8 + N (J, d, ǫ) 2 ) dǫ from which we conclude that

π ψ sup i∈J Y i ≤ 6 1/2 0 6 log (8 + N (J, d, ǫ) 2 ) dǫ
Using the fact that the ǫ-balls with center in I and intersecting J are necessarily contained in an (2ǫ)-ball with center in J, we also have

N (J, d, 2ǫ) ≤ N (I, d, ǫ)
This implies that

π ψ sup i∈J Y i ≤ 12 1 0 6 log (8 + N (I, d, ǫ) 2 ) dǫ
The end of the proof is now a direct consequence of the monotone convergence theorem with increasing series of finite subsets exhausting I. This ends the proof of the lemma.

Marginal inequalities

This section is mainly concerned with the proof of the theorem 4.1. This result is a more or less direct consequence of the following technical lemma of separate interest.

Lemma 4.6 Let M n := 0≤p≤n ∆ p be a real valued martingale with symmetric and independent increments (∆ n ) n≥0 . For any integer m ≥ 1, and any n ≥ 0, we have

E (|M n | m ) 1 m ≤ b(m) E [M ] m ′ /2 n 1 m ′ (82)
with the smallest even integer m ′ ≥ m, the bracket process

[M ] n := 0≤p≤n ∆ 2 p
and the collection of constants b(m) defined in [START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF]. In addition, for any m ≥ 2, we have

E (|M n | m ) 1 m ≤ b(m) (n + 1)   1 n + 1 0≤p≤n E |∆ p | m ′   1 m ′ (83)
Proof of theorem 4.1: We consider a collection of independent copies X ′ = (X ′i ) i≥1 of the random variables X = (X i ) i≥1 . We consider the martingale sequence M = (M i ) 1≤i≤N with symmetric and independent increments defined for any 1 ≤ j ≤ N by the following formula

M j := 1 √ N j i=1 f (X i ) -f (X ′i )
By construction, we have

V (X)(f ) = 1 √ N N i=1 f (X i ) -µ i (f ) = E (M N |X )
Comibing this conditioning property with the estimates provided in lemma 4.6, the proof of the first assertion is now easily completed. The Orlicz norm estimate (69) come from the fact that for any f ∈ Osc(E), we have

E(|V (X)(f )| 2m ) ≤ b(2m) 2m = E U 2m
for a Gaussian and centred random variable U , s.t. E(U 2 ) = 1. Using the comparison lemma, lemma 77, we find that

π ψ (V (X)(f )) ≤ π ψ (U ) = 8/3
Applying Kintchine's inequalities [START_REF] Rio | Local invariance principles and their applications to density estimation[END_REF], we prove that

E (|V (X)(f )| m ) 1 m ≤ b(m) E   1 N N i=1 f (X j ) -f (X ′j ) 2 m ′ /2   1/m ′ RR n°7677
By construction, we notice that for any f ∈ osc(E), and any p ≥ 2, we have

1 N N j=1 E f (X j ) -f (X ′j ) p ≤ 2σ(f ) 2
By the Rosenthal type inequality stated in theorem 2.5 in [START_REF] Johnson | Best Constants in Moment Inequalities for Linear Combinations of Independent and Exchangeable Random Variables[END_REF], for any sequence of nonnegative, independent and bounded random variables (Y i ) i≥1 , we have the rough estimate

E N i=1 Y p i 1/p ≤ 2p max   N i=1 E(Y i ), N i=1 E(Y p i ) 1/p  
for any p ≥ 1. If we take p = m ′ /2, and

Y i = 1 N f (X i ) -f (X ′i ) 2 we prove that E 1 N N i=1 f (X i ) -f (X ′i ) 2 m ′ /2 2/m ′ ≤ 4m max 2σ(f ) 2 , 1 N 1-2 m ′ 2σ(f ) 2 2/m ′
for any f ∈ osc(E). Using Stirling's approximation of factorials

√ 2πn n n e -n ≤ n! ≤ e √ 2πn n n e -n
for any p ≥ 1 we have

(2p) p /b(2p) 2p = 2 2p p p p!/(2p)! ≤ e p+1 ≤ 3 2p
and

(2p + 1) p+1/2 /b(2p + 1) 2p+1 = (2p + 1) p+1 2 p p!/(2p + 1)! ≤ e p+2 ≤ 3 2p+1
This implies that

m m/2 /b(m) m ≤ 3 m ⇒ √ m b(m) ≤ 3b(m) 2
for any m ≥ 1. This ends the proof of the theorem. Now, we come to the proof of the lemma. Proof of lemma 4.6: We prove the lemma by induction on the parameter n. The result is clearly satisfied for n = 0. Suppose the estimate ( 82) is true at rank (n -1). To prove the result at rank n, we use the binomial decomposition

(M n-1 + ∆ n ) 2m = 2m p=0 2m p M 2m-p n-1 (∆ n ) p RR n°7677
Using the symmetry condition, all the odd moments of ∆ n are null. Consequently, we find that

E (M n-1 + ∆ n ) 2m = m p=0 2m 2p E M 2(m-p) n-1 E ∆ 2p n
Using the induction hypothesis, we prove that the above expression is upper bounded by the quantity

m p=0 2m 2p 2 -(m-p) (2(m -p)) (m-p) E [M ] m-p n-1 E ∆ 2p n
To take the final step, we use the fact that

2m 2p 2 -(m-p) (2(m -p)) (m-p) = 2 -m (2m) m 2 -p (2p) p m p and (2p) p ≥ 2 p , to conclude that E (M n-1 + ∆ n ) 2m ≤ 2 -m (2m) m m p=0 m p E [M ] m-p n-1 E ∆ 2p n = 2 -m (2m) m E ([M ] m n )
For odd integers we use twice the Cauchy-Schwarz inequality to deduce that

E(|M n | 2m+1 ) 2 ≤ E(M 2m n ) E(M 2(m+1) n ) ≤ 2 -(2m+1) (2m) m (2(m + 1)) (m+1) E [M ] m+1 n 2m+1 m+1
We conclude that

E(|M n | 2m+1 ) ≤ 2 -(m+1/2) (2m + 1) (m+1) m + 1/2 E [M ] m+1 n 1- 1 2(m+1)
The proof of ( 82) is now completed. Now, we come to the proof of [START_REF] Rousset | On the control of an interacting particle approximation of Schroedinger ground states[END_REF]. For any m ′ ≥ 2 we have

  1 n + 1 0≤p≤n ∆ 2 p   m ′ /2 ≤ 1 n + 1 0≤p≤n E |∆ p | m ′
and therefore

E [M ] m ′ /2 n 1 m ′ ≤ (n + 1) 1/2   1 n + 1 0≤p≤n E |∆ p | m ′   1 m ′
This ends the proof of the lemma.

Maximal inequalities

The main goal of this section is to prove theorem 4.3. We begin with the basic symmetrization technique. We consider a collection of independent copies X ′ = (X ′i ) i≥1 of the random variables X = (X i ) i≥1 . Let ε = (ε i ) i≥1 constitute a sequence that is independent and identically distributed with

P (ε 1 = +1) = P (ε 1 = -1) = 1/2
We also consider the empirical random field sequences

V ǫ (X) := √ N m ε (X)
We also assume that (ǫ, X, X ′ ) are independent. We associate with the pairs (ǫ, X) and (ǫ,

X ′ ) the random measures m ǫ (X) = 1 N N i=1 ǫ i δ X i and m ǫ (X ′ ) = 1 N N i=1 ǫ i δ X ′i . We notice that m(X) -µ p F = sup f ∈F |m(X)(f ) -E(m(X ′ )(f ))| p ≤ E( m(X) -m(X ′ ) p F |X)
and in view of the symmetry of the random variables (f

(X i ) -f (X ′i )) i≥1 we have E( m(X) -m(X ′ ) p F ) = E( m ǫ (X) -m ǫ (X ′ ) p F ) from which we conclude that E ( V (X) p F ) ≤ 2 p E ( V ǫ (X) p F ) (84)
By using the Chernov-Hoeffding inequality for any x 1 , . . . , x N ∈ E, the empirical process

f -→ V ǫ (x)(f ) := √ N m ε (x)(f ) is sub-Gaussian for the norm f L2(m(x)) = m(x)(f 2 ) 1/2
. Namely, for any couple of functions f, g and any δ > 0 we have

E [V ǫ (x)(f ) -V ǫ (x)(g)] 2 = f -g 2 L2(m(x))
and by Hoeffding's inequality

P (|V ǫ (x)(f ) -V ǫ (x)(g)| ≥ δ) ≤ 2 e -1 2 δ 2 / f -g 2 L 2 (m(x)) If we set Z = Vǫ(x)(f ) √ 6 f L 2 (m(x))
2

, then we find that

E e Z -1 = ∞ 0 e t P (Z ≥ t) dt = ∞ 0 e t P |V ǫ (x)(f )| ≥ √ 6t f L2(m(x)) dt ≤ 2 ∞ 0 e t e -3t dt = 1 RR n°7677
from which we conclude that

π ψ (V ǫ (x)(f ) -V ǫ (x)(g)) ≤ √ 6 f -g L2(m(x))
Combining the maximal inequalities stated in lemma 4.5 and the conditioning property [START_REF] Pollard | Empirical processes : theory and applications[END_REF] we find that

π ψ ( V ǫ (X) F ) ≤ J(F ) with J(F ) ≤ 2 6 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ ≤ c I(F ) < ∞
for some finite universal constant c < ∞. Combining ( 84) with [START_REF] Metropolis | The Monte Carlo method[END_REF], this implies that π

ψ ( V (X) F ) ≤ 2 J(F )
This ends the proof of the theorem.

Cramér-Chernov inequalities

Some preliminary convex analysis

In this section, we present some basic Cramér-Chernov tools to derive quantitative concentration inequalities. We begin by recalling some preliminary convex analysis on Legendre-Fenchel transforms. We associate with any convex function

L : t ∈ Dom(L) → L(t) ∈ R +
defined in some domain Dom(L) ⊂ R + , with L(0) = 0, the Legendre-Fenchel transform L ⋆ defined by the variational formula

∀λ ≥ 0 L ⋆ (λ) := sup t∈Dom(L) (λt -L(t))
Note that L ⋆ is a convex increasing function with L ⋆ (0) = 0 and its inverse (L ⋆ ) -1 is a concave increasing function.

We let L A be the log-Laplace transform of a random variable A defined on some domain Dom(L A ) ⊂ R + by the formula

L A (t) := log E(e tA )
Hölder's inequality implies that L A is convex. Using the Cramér-Chernov-Chebychev inequality, we find that

log P (A ≥ λ) ≤ -L ⋆ A (λ) and P A ≥ (L ⋆ A ) -1 (x) ≤ e -x
for any λ ≥ 0 and any x ≥ 0.

The next lemma provides some key properties of Legendre-Fenchel transforms that will be used in several places in the further development of the lecture notes.

RR n°7677

Lemma 4.7

• For any convex functions (L 1 , L 2 ), such that

∀t ∈ Dom(L 2 ) L 1 (t) ≤ L 2 (t) and Dom(L 2 ) ⊂ Dom(L 1 )
we have

L ⋆ 2 ≤ L ⋆ 1 and (L ⋆ 1 ) -1 ≤ (L ⋆ 2 ) -1 • if we have ∀t ∈ v -1 Dom(L 2 ) = Dom(L 1 ) L 1 (t) = u L 2 (v t)
for some positive numbers (u, v) ∈ R 2 + , then we have

L ⋆ 1 (λ) = u L ⋆ 2 λ uv and (L ⋆ 1 ) -1 (x) = uv (L ⋆ 2 ) -1 x u
for any λ ≥ 0, and any ∀x ≥ 0.

• Let A be a random variable with a finite log-Laplace transform. For any a ∈ R, we have

L A (t) = -at + L A+a (t)
as well as

L ⋆ A (λ) = L ⋆ A+a (λ + a) and (L ⋆ A ) -1 (x) = -a + L ⋆ A+a -1 (x)
We illustrate this technical lemma with the detailed analysis of three convex increasing functions of current use in the further development of these notes

• L(t) = t 2 /(1 -t), t ∈ [0, 1[ • L 0 (t) := -t -1 2 log (1 -2t), t ∈ [0, 1/2[. • L 1 (t) := e t -1 -t
In the first situation, we readily check that

L ′ (t) = 1 (1 -t) 2 -1 and L ′′ (t) = 2 (1 -t) 3
An elementary manipulation yields that

L ⋆ (λ) = √ λ + 1 -1 2 and (L ⋆ ) -1 (x) = 1 + √ x 2 -1 = x + 2 √ x
In the second situation, we have

L ′ 0 (t) = 1 1 -2t -1 and L ′′ 0 (t) = 2 (1 -2t) 2
from which we find that

L ⋆ 0 (λ) = 1 2 (λ -log (1 + λ)) RR n°7677
We also notice that

L 0 (t) = t 2 p≥0 2 2 + p (2t) p ≤ L 0 (t) := t 2 1 -2t = 1 4 L(2t)
for every t ∈ [0, 1/2[ Using lemma 4.7, we prove that

L ⋆ 0 (λ) = 1 4 L ⋆ (2λ) ≤ L ⋆ 0 (λ) (L ⋆ 0 ) -1 (x) ≤ L ⋆ 0 -1 (x) = 1 2 (L ⋆ ) -1 (4x) = 2(x + √ x) (85)
In the third situation, we have

L ′ 1 (t) = e t -1 and L ′′ 1 (t) = e t
from which we conclude that

L ⋆ 1 (λ) = (1 + λ) log (1 + λ) -λ
On the other hand, using the fact that 2 × 3 p ≤ (p + 2)!, for any p ≥ 0, we prove that we have

L 1 (t) = t 2 2 p≥0 2 × 3 p (p + 2)! t 3 p ≤ L 1 (t) := t 2 2(1 -t/3) = 9 2 L t 3
for every t ∈ [0, 1/3[. This implies that L ⋆ 1 (λ) = 9 2 L ⋆ 2λ 3 ≤ L ⋆ 1 (λ)
and therefore

(L ⋆ 1 ) -1 (x) ≤ L ⋆ 1 -1 (x) = 3 2 (L ⋆ ) -1 2x 9 = x 3 + √ 2x (86)
Another crucial ingredient in the concentration analysis of the sum of two random variables is a deep technical lemma of J. Bretagnolle and E. Rio [START_REF] Rio | Local invariance principles and their applications to density estimation[END_REF]. In the further development of this chapter, we use this argument to obtain a large family of concentration inequalities that are asymptotically "almost sharp" in a wide variety of situations. Lemma 4.8 (J. Bretagnolle & E. Rio [START_REF] Rio | Local invariance principles and their applications to density estimation[END_REF]) For any pair of random variables A and B with finite log-Laplace transform in a neighborhood of 0, we have

∀x ≥ 0 (L ⋆ A+B ) -1 (x) ≤ (L ⋆ A ) -1 (x) + (L ⋆ B ) -1 (x) (87)
We also quote the following reverse type formulae that allows to turn most of the concentration inequalities developed in these notes into Bernstein style exponential inequalities. Lemma 4.9 For any (u, v) ∈ R + , we have

u (L ⋆ 0 ) -1 (x) + v (L ⋆ 1 ) -1 (x) ≤ L ⋆ a(u,v),b(u,v) -1 (x)
with the functions

a(u, v) := 2u + v 3 and b(u, v) := √ 2 u + v 2
and the Laplace function

L a,b (t) = b 2a 2 L (at) with L ⋆ a,b (λ) ≥ λ 2 2(b + λa) Proof:
Using the estimates ( 85) and ( 86) we prove that

u (L ⋆ 0 ) -1 (x) + v (L ⋆ 1 ) -1 (x) ≤ 2 u (x + √ x) + v x 3 + √ 2x = a(u, v) x + 2x b(u, v) with a(u, v) := 2u + v 3 and b(u, v) := √ 2 u + v 2
Now, using lemma 4.7, we observe that

a x + √ 2xb = L ⋆ a,b -1 (x) with L a,b (t) = b 2a 2 L (at) (88)
Finally, we have

L ⋆ (λ) = √ λ + 1 -1 2 ≥ (λ/2) 2 (1 + λ/2)
The r.h.s. inequality can be easily checked using the fact that

√ 1 + 2λ -1 2 = 2 (1 + λ) 2 -(1 + 2λ) (1 + λ) + √ 1 + 2λ ≥ λ 2 (1 + λ) ⇐ √ 1 + 2λ ≤ (1 + λ)
This implies that

L ⋆ a,b (λ) = b 2a 2 L ⋆ 2a b λ ≥ λ 2 2(b + λa)
This ends the proof of the lemma.

RR n°7677

Concentration inequalities

In this section, we investigate some elementary concentration inequalities for bounded and chi-square type random variables. We also apply these results to empirical processes associated with independent random variables. Proposition 4.1 Let A be a centred random variable such that A ≤ 1. If we set σ A = E(A 2 ) 1/2 , then for any t ≥ 0, we have

L A (t) ≤ σ 2 A L 1 (t) (89)
In addition, the probability of the following events

A ≤ σ 2 A (L ⋆ 1 ) -1 x σ 2 A ≤ x 3 + σ A √ 2x
is greater than 1e -x , for any x ≥ 0.

Proof:

To prove (89) we use the fact the decomposition

E e tA -1 -A = E (L 1 (tA)1 X<0 ) + E L 1 (tA)1 X∈[0,1] Since we have ∀x ≤ 0 L 1 (tx) ≤ (tx) 2 /2 and ∀x ∈ [0, 1] L 1 (tx) = x 2 n≥2 x n-2 t n /n! ≤ x 2 L 1 (t)
we conclude that

E e tA ≤ 1 + t 2 2 E(A 2 1 A<0 ) + L 1 (t)E A 2 1 A∈[0,1] ≤ 1 + L 1 (t)σ 2 A ≤ e L1(t)σ 2 A
Using lemma 4.7, we readily prove that

(L ⋆ A ) -1 (x) ≤ σ 2 A (L ⋆ 1 ) -1 x σ 2 A ≤ x 3 + σ A √ 2x
This ends the proof of the proposition.

Proposition 4.2 For any measurable function f , with 0 < osc(f) ≤ a, any N ≥ 1, and any t ≥ 0, we have

L √ N V (X)(f ) (t) ≤ N σ 2 (f /a) L 1 (at) (90)
In addition, the probability of the following events

V (X)(f ) ≤ a -1 σ 2 (f ) √ N (L ⋆ 1 ) -1 xa 2 N σ 2 (f ) ≤ xa 3 √ N + 2xσ(f ) 2 (91)
is greater than 1e -x , for any x ≥ 0.

RR n°7677

Proof:

Replacing f by f /a, there is no loss of generality to assume that a = 1. Using the same arguments as the ones we used in the proof of proposition 4.1, we find that log E e t(f

(X i )-µ i (f )) ≤ µ i f -µ i (f ) 2 L 1 (t)
from which we conclude that

L N (t) := log E e t √ N V (X)(f ) = N i=1 log E e t(f (X i )-µ i (f )) ≤ L N (t) := N σ 2 (f ) L 1 (t)
By lemma 4.7, we have

(L ⋆ N ) -1 (x) ≤ L ⋆ N -1 (x) = N σ 2 (f ) (L ⋆ 1 ) -1 x N σ 2 (f )
This ends the proof of the proposition.

E (|B| m ) 1/m ≤ b(2m) 2 c with c < ∞
for any m ≥ 1, with the finite constants b(m) defined in [START_REF] Del Moral | Particle Approximations of Lyapunov Exponents Connected to Schroedinger Operators and Feynman-Kac Semigroups[END_REF], we have

L B (t) ≤ ct + L 0 (ct) (92)
for any 0 ≤ ct < 1/2. In addition, the probability of the following events

B ≤ c 1 + (L ⋆ 0 ) -1 (x) ≤ c 1 + 2(x + √ x)
is greater than 1e -x , for any x ≥ 0.

Proof:

Replacing B by B/c, there is no loss of generality to assume that c = 1. We recall that b(2m) 2m = E(U 2m ) for every centred Gaussian random variable with

E(U 2 ) = 1 and ∀t ∈ [0, 1/2) m≥0 t m m! b(2m) 2m = 1 √ 1 -2t = E(exp tU 2 )
This implies that

E(exp {tB}) ≤ m≥0 t m m! b(2m) 2m = 1 √ 1 -2t
for any 0 ≤ t < 1/2. In other words, we have

L B-1 (t) := log E(exp {t(B -1)} ≤ L 0 (t) RR n°7677 and L B (t) = t + L B-1 (t) ≤ t + L 0 (t)
from which we conclude that

L ⋆ B (λ) = L ⋆ B-1 (λ -1) ⇒ (L ⋆ B ) -1 (x) = 1 + L ⋆ B-1 -1 (x) ≤ 1 + (L ⋆ 0 ) -1 (x)
This ends the proof of the proposition.

Remark 4. [START_REF] Carmona | Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses[END_REF] We end this section with some comments on the estimate [START_REF] Mahler | Multi-target Bayes filtering via first-order multi-target moments[END_REF].

Using the fact that b(m) ≤ b(2m) (see for instance [START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF]) we readily deduce from ( 70) that

E (|V (X)(f )| m ) 1 m ≤ 6 √ 2 b(2m) 2 σ(f )
for any m ≥ 1, and for any N s.t. 2σ 2 (f )N ≥ 1. Thus, if we set

B = |V (X)(f )| and c = 6 √ 2 σ(f )
in proposition 4.3, we prove that for any N s.t. 2σ 2 (f )N ≥ 1, and for any

0 ≤ t < 1/(12 √ 2 σ(f )) L |V (X)(f )| (t) ≤ 6 √ 2 σ(f ) t + L 0 (6 √ 2 σ(f ) t)
In addition, the probability of the following events

|V (X)(f )| ≤ 6 √ 2 σ(f ) 1 + (L ⋆ 0 ) -1 (x) ≤ 6 √ 2 σ(f ) 1 + 2(x + √ x)
is greater than 1e -x , for any x ≥ 0.

When N is chosen so that 2σ 2 (f )N ≥ 1, using (91) we improve the above inequality. Indeed, using this concentration inequality implies that for any f ∈ Osc(E), the probability of the following events

V (X)(f ) ≤ √ 2 σ(f ) x 3 + √ x (93)
is greater than 1e -x , for any x ≥ 0.

Perturbation analysis

This section is mainly concerned with the proof of theorem 4.2, and theorem 4.4. We recall that for any second order smooth function F on R d , for some d ≥ 1, F (m(X)(f )) stands for the random functionals defined by

f = (f i ) 1≤i≤d ∈ Osc(E) d → F (m(X)(f )) = F (m(X)(f 1 ), . . . , m(X)(f d )) ∈ R
Both results rely on the following second order decomposition of independent interest. Proposition 4.4 For any N ≥ 1, we have the decomposition

√ N [F (m(X)(f )) -F (µ(f ))] = V (X) [D µ (F )(f )] + 1 √ N R(X)(f )
with a first order functional D µ (F )(f ) defined in [START_REF] Méléard | Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations[END_REF], and a second order term R(X)(f ) such that

E (|R(X)(f )| m ) 1/m ≤ 1 2 b(2m) 2 ∇ 2 F f 1
for any m ≥ 1, with the parameter defined in [START_REF] Méléard | Probabilistic interpretation and approximations of some Boltzmann equations[END_REF].

Proof:

Using a Taylor first order expansion, we have

√ N [F (m(X)(f )) -F (µ(f ))] = ∇F (µ(f )) V (X)(f ) ⊤ + 1 √ N R(X)(f )
with the second order remainder term

R(X)(f ) := 1 0 (1 -t)V (X)(f ) ∇ 2 F (tm(X)(f ) + (1 -t)µ(f )) V (X)(f ) ⊤ dt We notice that ∇F (µ(f )) V (X)(f ) ⊤ = V (X) ∇F (µ(f )) f ⊤ and osc ∇F (µ(f )) f ⊤ ≤ d i=1 ∂F ∂u i (µ(f ))
It is also easily checked that

E (|R(X)(f )| m ) 1/m ≤ 1 2 d i,j=1 sup ν∈P(E) ∂ 2 F ∂u i ∂u j (ν(f )) E (|V (X)(f i )V (X)(f j )| m )
1/m and for any 1 ≤ i, j ≤ d, we have

E (|V (X)(f i )V (X)(f j )| m ) 1/m ≤ E V (X)(f j ) 2m 1/(2m) E V (X)(f j ) 2m 1/(2m) ≤ b(2m) 2
This ends the proof of the proposition.

We are now in position to prove theorem 4.2.

Proof of theorem 4.2:

We set N [F (m(X)(f )) -F (µ(f ))] = A + B with A = √ N V (X) [D µ (F )(f )] and B = R(X)(f ) RR n°7677
Combining proposition 4.2 with proposition 4.3, if we set

g = D µ (F )(f ) a = ∇F (µ(f )) 1 and c = 1 2 ∇ 2 F f 1 then we have L A (t) ≤ N σ 2 (g/a)L 1 (at) L B (t) = ct + L B-c (t) with L B-c (t) ≤ L 0 (ct)
On the other hand, we have

(L ⋆ A ) -1 (x) ≤ N a σ 2 (g/a) (L ⋆ 1 )
-1

x N σ 2 (g/a) and using the fact that

L B (t) = ct + L B-c (t)
we prove that

L ⋆ B (λ) = L ⋆ B-c (λ -c) ⇒ (L ⋆ B ) -1 (x) = c + L ⋆ B-c -1 (x) ≤ c 1 + (L ⋆ 0 ) -1 (x)
Using Bretagnolle-Rio's lemma, we find that

(L ⋆ A+B ) -1 (x) ≤ (L ⋆ A ) -1 (x) + (L ⋆ B ) -1 (x) ≤ N a -1 σ 2 (g) (L ⋆ 1 ) -1 xa 2 N σ 2 (g) + c 1 + (L ⋆ 0 ) -1 (x)
This ends the proof of the theorem. Now, we come to the proof of theorem 4.4.

Proof of theorem 4.4:

We consider the empirical processes

f ∈ F i → m(X)(f ) ∈ R
associated with d classes of functions F i , 1 ≤ i ≤ d, defined in section 4.1. We further assume that f i ∨ osc(f i ) ≤ 1, for any f i ∈ F i , and we set

π ψ ( V (X) F ) := sup 1≤i≤d π ψ ( V (X) Fi )
Using theorem 4.3, we have that

π ψ ( V (X) F ) ≤ 12 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ with N (F , ǫ) := sup 1≤i≤d N (F i , ǫ)
Using proposition 4.4, for any collection of functions

f = (f i ) 1≤i≤d ∈ F := d i=1 F i RR n°7677
we have

√ N sup f ∈F |F (m(X)(f )) -F (µ(f ))| ≤ ∇F µ ∞ d i=1 V (X) Fi + d 2 √ N ∇ 2 F ∞ d i=1 V (X) 2 Fi
If we set

A := ∇F µ ∞ d i=1 V (X) Fi
then we find that

π ψ (A) ≤ ∇F µ ∞ d i=1 π ψ V (X) Fi
By lemma 4.2, this implies that

E e tA ≤ (1 + tπ ψ (A)) e (tπ ψ (A)) 2 ≤ e at+ 1 2 t 2 b with b = 2a 2 and a = π ψ (A) ≤ ∇F µ ∞ d i=1 π ψ V (X) Fi Notice that L A-a (t) ≤ L(t) = 1 2 t 2 b
Recalling that

L ⋆ (λ) = λ 2 2b and (L ⋆ ) -1 (x) = √ 2bx
we conclude that

(L ⋆ A ) -1 (x) = a + L ⋆ A-a -1 (x) ≤ a + √ 2bx = π ψ (A) 1 + 2 √ x
Now, we come to the analysis of the second order term defined by

B = d 2 √ N ∇ 2 F ∞ d i=1 V (X) 2 Fi
Using the inequality

E(B m ) ≤ β m d m-1 d i=1 E V (X)
2m Fi

RR n°7677

with

β := d 2 √ N ∇ 2 F ∞
Combining lemma 4.2 with theorem 4.3, we conclude that

E(B m ) ≤ m! β d π ψ ( V (X) F ) 2 m If we set b := β d π ψ ( V (X) F ) 2
then we have that

E e tB ≤ m≥0 (bt) m = 1 1 -bt = e bt × e 2L0(bt/2)
for any 0 ≤ t < 1/b with the convex increasing function L 0 introduced on page 70, so that 2L 0 (bt/2) = -btlog (1bt)

Using lemma 4.7, we prove that

L B-b (t) ≤ 2L 0 (bt/2)
and

(L ⋆ B ) -1 (x) = b + L ⋆ B-b -1 (x) ≤ b 1 + (L ⋆ 0 ) -1 x 2 = 1 2 √ N ∇ 2 F ∞ (d π ψ ( V (X) F )) 2 1 + (L ⋆ 0 ) -1 x 2
Finally, using the Bretagnolle-Rio's lemma, we prove that

L ⋆ A+B -1 (x) ≤ dπ ψ ( V (X) F ) ∇F µ ∞ (1 + 2 √ x) + 1 2 √ N ∇ 2 F ∞ (d π ψ ( V (X) F )) 1 + (L ⋆ 0 ) -1 x 2
This ends the proof of the theorem 4.4.

5 Interacting empirical processes

Introduction

This short chapter is concerned with the concentration analysis of sequences of empirical processes associated with conditionally independent random variables.

In preparation for the work in chapter 6 on the collection of Feynman-Kac particle models introduced in section 1.4, we consider a general class of interaction particle processes with non necessarily mean field type dependency.

Firstly, we analyze the concentration properties of integrals of local sampling error sequences, with general random but predictable test functions. These results will be used to analyze the concentration properties of the first order fluctuation terms of the particle models.

We also present a stochastic perturbation technique to analyze the second order type decompositions. We consider finite marginal models and empirical processes. We close the chapter with the analysis of the covering numbers and the entropy parameters of linear transformation of classes of functions.

We end this introductory section, with the precise description of the main mathematical objects we shall analyze in the further development of the chapter.

We let

X (N ) n = (X (N,i) n
) 1≤i≤N be a Markov chain on some product state spaces E N n , for some N ≥ 1. We also let G N n be the increasing σ-field generated by the random sequence (X (N ) p ) 0≤p≤n . We further assume that (X (N,i) n

) 1≤i≤N are conditionally independent, given G N n-1 . As traditionally, when there is no possible confusion, we simplify notation and suppress the index (.) (N ) so that we write

(X n , X i n , G n ) instead of (X (N ) n , X (N,i) n , G N n ).
In this simplified notation, we also denote by µ i n the conditional distribution of the random state X i n given the G n-1 ; that is, we have that

µ i n = Law(X i n | G n-1 )
Notice that the conditional distributions

µ n := 1 N N i=1 µ i n
represent the local conditional mean of the occupation measures

m(X n ) := 1 N N i=1 δ X i n
At this level of generality, we cannot obtain any kind of concentration properties for the deviations of the occupation measures m(X n ) around some deterministic limiting value.

In chapter 6, dedicated to particle approximations of Feynman-Kac measures η n , we shall deal with mean field type random measures µ i n , in the sense that the randomness only depends on the location of the random state X i n-1 and on the current occupation measure m(X n-1 ). In this situation, the fluctuation of m(X n ) around the limiting deterministic measures η n will be expressed in terms of second order Taylor's type expansions w.r.t. the local sampling errors

V (X p ) = √ N (m(X p ) -µ p )
from the origin p = 0, up to the current time p = n.

The first order terms will be expressed in terms of integral formulae of predictable functions f p w.r.t. the local sampling error measures V (X p ). These stochastic first order expansions are defined below. Definition 5.1 For any sequence of G n-1 -measurable random function f n ∈ Osc(E n ), and any numbers a n ∈ R + , we set

V n (X)(f ) = n p=0 a p V (X p )(f p ) (94)
For any G n-1 -measurable random function f n ∈ Osc(E n ), we have

E (V (X n )(f n ) |G n-1 ) = 0 E V (X n )(f n ) 2 |G n-1 = σ N n (f n ) 2 := 1 N N i=1 µ i n f n -µ i n (f n ) 2
We also assume that we have an almost sure estimate

sup N ≥1 σ N n (f n ) 2 ≤ σ 2 n for some positive constant σ 2 n ≤ 1. (95)

Finite marginal models

We will now derive a quantitative contraction inequality for the general random fields models of the following form

W n (X)(f ) = V n (X)(f ) + 1 √ N R n (X)(f ) (96)
with V n (X)(f ) defined in [START_REF] Whitley | The island model genetic algorithm: On separability, population size and convergence[END_REF], and a second order term such that

E (|R n (X)(f )| m ) 1/m ≤ b(2m) 2 c n
for any m ≥ 1, for some finite constant c n < ∞ whose values only depend on the parameter n.

For a null remainder term R n (X)(f ) = 0, these concentration properties are easily derived using proposition 4.2.

Proposition 5.1 We let V n (X)(f ) be the random field sequence defined in [START_REF] Whitley | The island model genetic algorithm: On separability, population size and convergence[END_REF]. For any t ≥ 0, we have that

L √ N Vn(X)(f ) (t) ≤ N σ 2 n L 1 (ta ⋆ n )
with the parameters

σ 2 n := 0≤p≤n σ 2 p and a ⋆ n := max 0≤p≤n a p
In addition, the probability of the following events

V n (X)(f ) ≤ √ N a ⋆ n σ 2 n (L ⋆ 1 ) -1 x N σ 2 n ≤ a ⋆ n x 3 √ N + 2σ 2 n x
is greater than 1e -x , for any x ≥ 0.

RR n°7677

Proof: By proposition 4.2, we have

E e t √ N Vn(X)(f ) |G n-1 = e t √ NVn-1(X)(f ) E e (tan) √ N V (Xn)(fn) |G n-1 with log E e (tan) √ N V (Xn)(fn) |G n-1 ≤ N σ 2 n L 1 (ta ⋆ n ) This clearly implies that L √ N Vn(X)(f ) (t) ≤ L 1 (t) := N σ 2 n L 1 (ta ⋆ n )
Using lemma 4.7, we conclude that

L ⋆ √ N Vn(X)(f ) -1 (x) ≤ L ⋆ 1 -1 (x) = N a ⋆ n σ 2 n (L ⋆ 1 ) -1 x N σ 2 n
The last assertion is a direct consequence of [START_REF] Talagrand | New concentration inequalities in product spaces[END_REF]. This ends the proof of the proposition.

Theorem 5.1 We let W n (X)(f ) be the random field sequence defined in (96) In this situation, the probability of the events

√ N W n (X)(f ) ≤ c n 1 + (L ⋆ 0 ) -1 (x) + N a ⋆ n σ 2 n (L ⋆ 1 ) -1
x N σ 2 n is greater than 1e -x , for any x ≥ 0. In the above display, σ n stands for the variance parameter definition in proposition 5.1

Proof: We set √ N W n (X)(f ) = A n + B n , with A n = √ N V n (X)(f ) and B n = R n (X)(f )
By proposition 4.3 and proposition 5.1, we have

L An (t) ≤ L An (t) := N σ 2 n (f ) L 1 (ta ⋆ n ) and L Bn-cn (t) ≤ L Bn-cn (t) := L 0 (c n t)
We recall that

L Bn (t) = c n t + L Bn-cn (t) ⇒ L ⋆ Bn (λ) = L ⋆ Bn-cn (λ -c n )
Using lemma 4.7, we also have that

L ⋆ Bn -1 (x) = c n + L ⋆ Bn-cn -1 (x) ≤ c n + L ⋆ Bn-cn -1 (x) = c n 1 + (L ⋆ 0 ) -1 (x)
In the same vein, arguing as in the end of the proof of proposition 5.1, we have

L ⋆ √ N Vn(X)(f ) -1 (x) ≤ N a ⋆ n σ 2 n (L ⋆ 1 ) -1 x N σ 2 n
The end of the proof is now a direct consequence of the Bretagnolle-Rio's lemma. This ends the proof of the theorem.

Empirical processes

We let V n (X) be the random field sequence defined in [START_REF] Whitley | The island model genetic algorithm: On separability, population size and convergence[END_REF], and we consider a sequence of classes of G n-1 -measurable random functions F n , such that

f n ∨ osc(f n ) ≤ 1, for any f n ∈ F n . Definition 5.2 For any f = (f n ) n≥0 ∈ F := (F n ) n≥0 , and any sequence of numbers a = (a n ) n≥0 ∈ R N + , we set V n (X)(f ) = n p=0 a p V (X p )(f p ) and V n (X) F = sup f ∈F |V n (X)(f )|
We further assume that for any n ≥ 0, and any ǫ > 0, we have an almost sure estimate

N (F n , ǫ) ≤ N n (ǫ) (97)
for some non increasing function N n (ǫ) such that

b n := 12 2 2 0 log (8 + N n (ǫ) 2 ) dǫ < ∞
In this situation, we have

π ψ ( V n (X) F ) ≤ n p=0 a p π ψ V (X p ) Fp
Using theorem 4.3, given G n-1 we have the almost sure upper bound

π ψ V (X p ) Fp ≤ 12 2 2 0 log (8 + N (F p , ǫ) 2 ) dǫ ≤ b p
Combining lemma 4.1 and lemma 4.2, we readily prove the following theorem.

Theorem 5.2 For any classes of G n-1 -measurable random functions F n satisfying the entropy condition (97), we have

π ψ ( V n (X) F ) ≤ c n := n p=0 a p b p
In particular, the probability of the events

V n (X) F ≤ c n x + log 2
is greater than 1e -x , for any x ≥ 0.

RR n°7677

Next, we consider classes of non random functions F = (F n ) n≥0 . We further assume that f n ∨ osc(f n ) ≤ 1, for any f n ∈ F n , and

I 1 (F ) := 12 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ < ∞ with N (F , ǫ) = sup n≥0 N (F n , ǫ) < ∞ Theorem 5.3 We let W n (X)(f ), f ∈ F , be the random field sequence defined by W n (X)(f ) = V n (X)(f ) + 1 √ N R n (X)(f )
with a second order term such that

E sup f ∈F |R n (X)(f )| m ≤ m! c m n
for any m ≥ 1, for some finite constant c n < ∞ whose values only depend on the parameter n. In this situation, the probability of the events

W n (X) F ≤ n p=0 a p I 1 (F ) 1 + 2 √ x + c n √ N 1 + (L ⋆ 0 ) -1 x 2
is greater than 1e -x , for any x ≥ 0.

Proof:

We

set √ N W n (X) F ≤ A n + B n , with A n = √ N V n (X) F and B n = sup f ∈F |R n (X)(f )| Using the fact that sup f ∈F |V n (X)(f )| ≤ n p=0 a p V (X p ) Fp by lemma 4.1, we have π ψ ( V n (X) F ) ≤ n p=0 a p π ψ V (X p ) Fp
Using theorem 4.3, we also have that

π ψ V (X p ) Fp ≤ I 1 (F ) := 12 2 2 0 log (8 + N (F , ǫ) 2 ) dǫ with N (F , ǫ) = sup n≥0 N (F n , ǫ) RR n°7677
This implies that

π ψ (A n ) ≤ a n √ N I 1 (F ) with a n := n p=0 a p
By lemma 4.2, we have

E e tAn ≤ (1 + tπ ψ (A n )) e (tπ ψ (An)) 2 ≤ e αnt+ 1 2 t 2 βn with β n = 2α 2 n and α n = π ψ (A n ) Notice that L An-αn (t) ≤ L n (t) := 1 2 t 2 β n Recalling that L ⋆ n (λ) = λ 2 2β n and (L ⋆ n ) -1 (x) = 2β n x we conclude that L ⋆ An -1 (x) = α n + L ⋆ An-αn -1 (x) ≤ α n + 2β n x = π ψ (A n ) 1 + 2 √ x
On the other hand, under our assumption, we also have that

E e tBn ≤ m≥0 (c n t) m = 1 1 -c n t = e cnt × e 2L0(cnt/2)
for any 0 ≤ t < 1/c n with the convex increasing function L 0 introduced on page 70, so that 2L 0 (c n t/2) = -c n tlog (1c n t)

Using lemma 4.7, we conclude that

L Bn-cn (t) ≤ 2L 0 (c n t/2)
and

L ⋆ Bn -1 (x) = c n + L ⋆ Bn-cn -1 (x) ≤ c n 1 + (L ⋆ 0 ) -1 x 2
The end of the proof is now a direct consequence of the Bretagnolle-Rio's lemma.

RR n°7677

Covering numbers and entropy methods

In this final section, we derive some properties of covering numbers for some classes of functions. These two results are the key to derive uniform concentration inequalities w.r.t. the time parameter for Feynman-Kac particle models. This subject is investigated in chapter 6.

We let (E n , E n ) n=0,1 be a pair of measurable state spaces, and F be a separable collection of measurable functions f :

E 1 → R such that f ≤ 1 and osc(f ) ≤ 1.
We consider a Markov transition M (x 0 , dx 1 ) from E 0 into E 1 , a probability measure µ on E 0 , and a function G from E 0 into [0, 1] . We associate with these objects the class of functions

G • M (F ) = {G M (f ) : f ∈ F } and G • (M -µM )(F ) = {G [M (f ) -µM (f )] : f ∈ F } Lemma 5.1 For any ǫ > 0, we have N [G • M (F ), ǫ] ≤ N (F , ǫ)
Proof:

For any probability measure η on E 0 , we let {f 1 , . . . , f nǫ } be the centers of n ǫ = N (F , L 2 (η), ǫ) L 2 (η)-balls of radius at most ǫ covering F . For any f ∈ F , there exists some

1 ≤ i ≤ n ǫ such that η [G(f -f i )] 2 1/2 ≤ η [(f -f i )] 2 1/2 ≤ ǫ This implies that N (G • F , L 2 (η), ǫ) ≤ N (F , L 2 (η), ǫ)
In much the same way, we let {f 1 , . . . , f nǫ } be the n ǫ = N (F , L 2 (ηM ), ǫ) centers of L 2 (ηM )-balls of radius at most ǫ covering F . In this situation, for any f ∈ F , there exists some

1 ≤ i ≤ n ǫ such that η [(M (f ) -M (f i ))] 2 1/2 ≤ ηM [(f -f i )] 2 1/2 ≤ ǫ This implies that N (M (F ), L 2 (η), ǫ) ≤ N (F , L 2 (ηM ), ǫ)
This ends the proof of the lemma.

One of the simplest way to control the covering numbers of the second class of functions is to assume that M satisfies the following condition M (x, dy) ≥ δν(dy), for any x ∈ E 0 , and for some measure ν, and some δ ∈]0, 1[. Indeed, in this situation we observe that

M δ (x, dy) = M (x, dy) -δν(dy) 1 -δ RR n°7677
6 Feynman-Kac particle processes

Introduction

In this chapter, we investigate the concentration properties of the collection of Feynman-Kac particle measures introduced in section 1.4, in terms of the contraction parameters τ k,l (n) and τ k,l (m) introduced in definition 3.3, and in corollary 3.1.

In the first section, section 6.2, we present some basic first order decompositions of the Boltzmann-Gibbs transformation associated with some regular potential function.

In section 6.3, we combine the semigroup techniques developed in chapter 3, with a stochastic perturbation analysis to derive first order integral expansions in terms of local random fields and and Feynman-Kac transport operators.

In section 6.4, we combine these key formulae with the concentration analysis of interacting empirical processes developed in chapter 5. We derive quantitative concentration estimates for finite marginal models, as well as for empirical processes w.r.t. some classes of functions. The final two sections, section 6.5, and section 6.6, are devoted respectively to particle free energy models, and backward particle Markov models.

First order expansions

For any positive potential function G, any measures µ and ν, and any function f on E, we have

[Ψ G (µ) -Ψ G (ν)] (f ) = 1 µ(Gν ) (µ -ν)(d ν Ψ G (f )) = 1 -1 µ(Gν ) (µ -ν)(G ν ) (µ -ν)(d ν Ψ G (f )) (98)
with the functions

d ν Ψ G (f ) := G ν (f -Ψ G (ν)(f )) and G ν := G/ν(G) (99) Notice that |[Ψ G (µ) -Ψ G (ν)] (f )| ≤ g |(µ -ν)(d ν Ψ G (f ))| and d ν Ψ G (f ) ≤ g osc(f ) with g := sup x,y (G(x)/G(y))
It is also important to observe that

|[Ψ G (µ) -Ψ G (ν)] (f )| ≤ 1 µ(G ′ ) |(µ -ν)(d ′ ν Ψ G (f ))| ≤ g |(µ -ν)(d ′ ν Ψ G (f ))| with the integral operator d ′ ν Ψ G from Osc(E) into itself defined by d ′ ν Ψ G (f ) := G ′ (f -Ψ G (ν)(f )
) and G ′ := G/ G Using lemma 5.2, we readily prove the following lemma.

RR n°7677

Lemma 6.1 We let F be separable collection of measurable functions f : E ′ → R on some possibly different state space E ′ , and such that f ≤ 1 and osc(f ) ≤ 1. For any Markov transition M from E into E ′ , we set

d ′ ν Ψ G M (F ) := {d ′ ν Ψ G (M (f )) : f ∈ F } In this situation, we have the uniform estimate sup ν∈P(E) N [d ′ ν Ψ G M (F ), 2ǫβ(M )] ≤ N (F , ǫ) (100)

A stochastic perturbation analysis

Mean field particle models can be thought as a stochastic perturbation technique for solving nonlinear measure valued equations of the form

η n = Φ n (η n-1 )
The random perturbation term is encapsulated into the sequence of local random sampling errors (V N n ) n≥0 given by the local perturbation equations

η N n = Φ n η N n-1 + 1 √ N V N n
One natural way to control the fluctuations and the concentration properties of the particle measures

(η N n , γ N n ) around their limiting values (η n , γ n ) is to express the random fields (W γ,N n , W η,N n ) defined by γ N n = γ n + 1 √ N W γ,N n η N n = η n + 1 √ N W η,N
n in terms of the empirical random fields (V N n ) n≥0 . As shown in [START_REF] Ledoux | The concentration of measure phenomenon[END_REF], it is important to recall that the local sampling random fields models V N n belong to the class of empirical processes we analyzed in section 4.1. The stochastic analysis developed in chapter 4 applies directly to these models. For instance, using theorem 4.1 we have the quantitative almost sure estimate of the amplitude of the stochastic perturbations

E V N n (f ) m G N n-1 1/m ≤ b(m) (101)
for any m ≥ 1 and any test function f ∈ Osc(E n )

The first order expansions presented in the further development of this section will be expressed in terms of the random functions d N p,n (f ) and G N p,n , and the first order functions d p,n (f ) defined below.

G N p,n := G p,n Φ p (η N p-1 )(G p,n ) d N p,n (f ) := d Φp(η N p-1 ) Ψ Gp,n (P p,n (f )) RR n°7677 and d ′N p,n (f ) := d ′ Φp(η N p-1 ) Ψ Gp,n (P p,n (f )) Notice that G N p,n ≤ g p,n and d p,n (f ) ∨ d N p,n (f ) ≤ g p,n β(P p,n )
as well as

d ′N p,n (f ) ≤ β(P p,n ) and osc d ′N p,n (f ) ≤ 2β(P p,n )
As promised, the next theorem presents some key first order decompositions which are the progenitors for our other results. Further details on these expansions and their use in the bias and the fluctuation analysis of Feynman-Kac particle models can be found [START_REF] Moral | Non Linear Filtering: Interacting Particle Solution[END_REF][START_REF] Moral | Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-Linear Filtering[END_REF][START_REF] Moral | Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications[END_REF][START_REF] Del Moral | Sequential Monte Carlo & Genetic particle models[END_REF]. Theorem 6.1 For any 0 ≤ p ≤ n, and any function f on E n , we have the decomposition

W η,N n (f ) = n p=0 1 η N p (G N p,n ) V N p (d N p,n (f )) (102)
and the L m -mean error estimates

E W η,N n (f ) m 1/m ≤ 2 b(m) τ 1,1 (n) (103)
with the parameter τ 1,1 (n) defined in [START_REF] Makrini | Diffusion Monte Carlo method: Numerical analysis in a simple case[END_REF].

In addition, we have

W η,N n (f ) = n p=0 V N p [d p,n (f )] + 1 √ N R N n (f ) (104)
with a second order remainder term

R N n (f ) := - n-1 p=0 1 η N p (G p ) W η,N p (G p ) W η,N p [d p,n (f )] such that sup f ∈Osc(En) E R N n (f ) m 1/m ≤ 4 b(2m) 2 τ 2,1 (n) (105)
with the parameter τ 2,1 (n) defined in [START_REF] Makrini | Diffusion Monte Carlo method: Numerical analysis in a simple case[END_REF].

Proof:

The proof of (102) is based on the telescoping sum decomposition

η N n -η n = n p=0 Φ p,n (η N p ) -Φ p,n Φ p (η N p-1 )
Recalling that Φ p,n (µ) = Ψ Gp,n (µ)P p,n

RR n°7677

Finally, we observe that

d ηp-1 Ψ Gp-1 (M p (d p,n (f ))) = G p-1 η p-1 (G p-1 ) (M p (d p,n (f )) -η p (d p,n (f ))) = G p-1 η p-1 (G p-1 ) M p (d p,n (f )) = Q p-1,p (d p,n (f )) = Q p-1,n (f -η n (f )) = d p-1,n (f )
This ends the proof of (104). Combining (102) with the almost sure estimates

E V N p (d N p,n (f n )) m G N p-1 1/m ≤ 2b(m) G N p,n β(P p,n )
for any m ≥ 1, we easily prove (103). Using the fact that

1 inf x G N p,n (x) E V N p (G N p,n ) m G N p-1 1/m ≤ 2 b(m) g p,n and
E V N p d N p,n (f ) m G N p-1 1/m ≤ 2 b(m) g p,n β(P p,n )
we prove (105). This ends the proof of the theorem.

Concentration inequalities

Finite marginal models

In section 3.6, dedicated to the variance analysis of the local sampling models, we have seen that the empirical random fields V N n satisfy the regularity conditions of the general interacting empirical process models V (X n ) presented in section 5.1.

To be more precise, we have the formulae

n p=0 V N p [d p,n (f )] = n p=0 a p V N p [δ p,n (f )]
with the functions

δ p,n (f ) = d p,n (f )/a p ∈ Osc(E p ) ∩ B 1 (E p )
for any finite constants

a p ≥ 2 sup 0≤p≤n (g p,n β(P p,n ))
Therefore, if we fix the final time horizon n, with some slightly abusive notation we have the formulae

n p=0 V N p [d p,n (f )] = n p=0 α p V (X p ) [f p ] RR n°7677
with

X p = ξ p = ξ i p 1≤i≤N α p = sup 0≤q≤n a q = a ⋆ n and f p = δ p,n (f )/a ⋆ n
We also notice that

E V N p [δ p,n (f )] 2 G N p-1 ≤ 1 (a ⋆ n ) 2 σ 2 p osc(d p,n (f )) 2 ≤ 4 (a ⋆ n ) 2 σ 2 p d p,n (f ) 2
and therefore

E V N p [δ p,n (f )] 2 G N p-1 ≤ 4 (a ⋆ n ) 2 σ 2 p g 2 p,n β(P p,n ) 2
with the uniform local variance parameters σ 2 p defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF]. This shows that the regularity condition stated in (95) is met by replacing the parameters σ p in the variance formula (95) by the constants 2σ p g p,n β(P p,n )/a ⋆ n , with the uniform local variance parameters σ p defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF].

Using theorem 5.1, we easily prove the following exponential concentration property. Theorem 6.2 ( [START_REF] Moral | Concentration inequalities for mean field particle models[END_REF]) For any n ≥ 0, any f ∈ Osc(E n ), and any N ≥ 1, the probability of the event

η N n -η n (f ) ≤ 4τ 2,1 (n) N 1 + (L ⋆ 0 ) -1 (x) + 2b n σ 2 n (L ⋆ 1 )
-1

x N σ 2 n is greater than 1e -x , for any x ≥ 0, with

σ 2 n := 1 b 2 n 0≤p≤n g 2 p,n β(P p,n ) 2 σ 2 p
for any choice of b n ≥ κ(n). In the above display, τ 2,1 (n) and κ(n) stands for the parameter defined in [START_REF] Makrini | Diffusion Monte Carlo method: Numerical analysis in a simple case[END_REF], and σ n is the uniform local variance parameter defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF].

We illustrate the impact of this theorem with two applications. The first one is concerned with regular and stable Feynman-Kac models satisfying the regularity conditions presented in section 3.4. The second one is concerned with the concentration properties of the genealogical tree based models developed in section 14.

In the first situation, combining corollary 3.1, with the estimates ( 85) and (86) we prove the following uniform concentration inequalities w.r.t. the time horizon. Corollary 6.1 We assume that one of the regularity conditions H m (G, M) stated in section 3.4.1 is met for some m ≥ 0, and we set

p m (x) = 4τ 2,1 (m) 1 + 2(x + √ x) + 2 3 κ(m) x RR n°7677 and q m (x) = 8σ 2 τ 2,2 (m) x with σ 2 = sup n≥0 σ 2 n
In the above displayed formula, τ 2,2 (m) and κ(m) stands for the parameters defined in corollary 3.1, and σ n is the uniform local variance parameter defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF].

In this situation, for any n ≥ 0, any f ∈ Osc(E n ), any N ≥ 1, and for any x ≥ 0, the probability of the event

η N n -η n (f ) ≤ 1 N p m (x) + 1 √ N q m (x)
is greater than 1e -x .

In the same vein, using the estimates ( 85) and ( 86), concentration inequalities for genealogical tree models can be derived easily using the estimates (58). Corollary 6.2 We let η N n be the occupation measure of the genealogical tree model presented in [START_REF] Johansen | Sequential Monte Carlo Samplers for Rare Events[END_REF]. We also set σ 2 = sup n≥0 σ 2 n , the supremum of is the uniform local variance parameters σ 2 n defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF], and

p n,m (x) = 4(χ m g m ) 2 1 + 2(x + √ x) + 2 3 χ m g m (n + 1) x and q m (x) = (χ m g m ) √ 8σ 2
x In this situation, for any n ≥ 0, any f n ∈ Osc(E n ), and any N ≥ 1, the probability of the event

η N n -Q n (f n ) ≤ n + 1 N p n,m (x) + n + 1 N q m (x)
is greater than 1e -x , for any x ≥ 0.

Empirical processes

The main aim of this section is to derive concentration inequalities for particle empirical processes. Several consequences of this general theorem are also discussed, including uniform estimates w.r.t. the time parameter, and concentration properties of genealogical particle processes.

Theorem 6.3 We let F n be a separable collection of measurable functions f n on E n , such that f n ≤ 1, osc(f n ) ≤ 1, with finite entropy I(F n ) < ∞. In particular, for any n ≥ 0, and any N ≥ 1, the probability of the following event sup

f ∈Fn η N n (f ) -η n (f ) ≤ c Fn √ N τ 1,1 (n) x + log 2
is greater than 1e -x , for any x ≥ 0.

RR n°7677

To begin with, we mention that

γ N n (1) := γ N n (1)/γ n (1) = 0≤p<n η N p (G p ) = 1 + 1 √ N W γ,N n (1)
For more general functions we also observe that for any function f on E n , s.t. η n (f ) = 1, we have the decompositions

W γ,N n (f ) = √ N 1 + 1 √ N W γ,N n (1) η n (f ) + 1 √ N W η,N n (f ) -η n (f ) = √ N 1 + 1 √ N W γ,N n (1) 1 + 1 √ N W η,N n (f ) -1
We readily deduce the following second order decompositions of the fluctuation errors In the first part of this section, we provide some key decompositions of W γ,N n in terms of the local sampling errors V N n , as well as a pivotal exponential formula connecting the fluctuations of the particle free energies in terms of the fluctuations of the potential empirical mean values.

In the second part of the section, we derive first order expansions, and logarithmic concentration inequalities for particle free energy ratios γ N n (1) = γ N n (1)/γ n (1).

Some key decomposition formulae

This section is mainly concerned with the proof of the following decomposition theorem.

Theorem 6.4 For any 0 ≤ p ≤ n, and any function f on E n , we have the decompositions

W γ,N n (f ) = n p=0 γ N p (1) V N p (Q p,n (f )) (107) W γ,N n (f ) = n p=0 γ N p (1) V N p (Q p,n (f )) (108)
with the normalized Feynman-Kac semigroup

Q p,n (f ) = Q p,n (f )/η p Q p,n (1)
In addition, we have the exponential formulae In much the same way, we have E R N,2 n r 1/r ≤ 0≤q≤p<n g q,p E V N q (G N q,p )

2r 1/2r E V N q d N q,p (G p ) 2r 1/2r
and using (101), we prove that E R N,2 n r 1/r ≤ 8b(2r) 2 0≤q≤p<n g p g 3 q,p β(P q,p )

is greater than 1e -x , for any x ≥ 0, for some function ρ N n such that ρ N n (x) → N →∞ 0

In this case, the probability of event -1e -nρ N n (x) ≤ γ N n (1) -1 ≤ e nρ N n (x) -1 is greater than 1 -2e -x , for any x ≥ 0. Choosing N large enough so that ρ N n (x) ≤ 1/n we have

-2nρ N n (x) ≤ -1 -e -nρ N n (x)
and e nρ N n (x) -1 ≤ 2nρ N n (x)

from which we conclude that the probability of event

P γ N n (1) -1 ≤ 2n ρ N n (x) ≥ 1 -2e -x
Now, we come to the proof of the theorem.

Proof of theorem 6.5:

We use the same line of arguments as the ones we used in section 6.4.1. Firstly, we observe that h N q,n ≤ q≤p<n d N q,p (G p ) ≤ q≤p<n g q,p osc(P q,p (G p )) ≤ 2 q≤p<n g q,p g p β(P q,p ) = c q,n /2 and osc(h N q,n ) ≤ c q,n . Now, we use the following decompositions 0≤q<n V N q h N q,n = a ⋆ n 0≤q<n

V N q δ N q,n with the G N q-1 -measurable functions δ N q,n = h N q,n /a ⋆ n ∈ Osc(E q ) ∩ B 1 (E q )

and for any constant a ⋆ n ≥ sup 0≤q≤n c q,n . On the other hand, we have the almost sure variance estimate

E V N q δ N q,n 2 G N q-1 ≤ σ 2
q osc(h N q,n ) 2 /a ⋆2 n ≤ σ 2 q c 2 q,n /a ⋆2 n from which we conclude that

E V N q δ N q,n 2 ≤ σ 2 q c 2 q,n /a ⋆2 n
This shows that the regularity condition stated in (95) is met by replacing the parameters σ q in the variance formula (95) by the constants σ q c q,n /a ⋆ n , with the uniform local variance parameters σ p defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF].

The end of the proof is now a direct consequence of theorem 5.1. This ends the proof of the theorem. In the above displayed formula, τ 2,2 (m) and κ(m) stands for the parameters defined in corollary 3.1, and σ n is the uniform local variance parameter defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF]. In this situation, for any N ≥ 1, and any ǫ ∈ {+1, -1}, the probability of each of the following events

ǫ n log γ N n (1) ≤ 1 N p m (x) + 1 √ N q m (x)
is greater than 1e -x , for any x ≥ 0,

Proof:

Under condition H m (G, M), we have r(n)/n ≤ (4gτ

In much the same way, we prove that ς ⋆ n ≤ 4β m g m+1 . The end of the proof is now a consequence of the estimates ( 85) and [START_REF] Talagrand | New concentration inequalities in product spaces[END_REF]. This ends the proof of the corollary.

Backward particle Markov models

This section is concerned with the concentration properties of the backward Markov particle measures defined in [START_REF] Dawson | Stochastic evolution equations and related measure processes[END_REF]. Without further mention, we assume that the Markov transitions M n satisfy the regularity condition [START_REF] Fr | Particle approximations of a class of branching distribution flows arising in multi-target tracking HAL[END_REF], and we consider the random fields defined below.

We check this claim, using the fact that The main reason for introducing these integral operators comes from the following integral transport properties.

and

η N p D N p,n (f n ) η N p D N p,n (1)
= Ψ Gp,n η N p L N p,n (f n ) (114) = Ψ Gp+1,n Φ p+1 (η N p ) L N p+1,n (f n ) (115)
Proof:

We check (113) using the fact that This ends the proof of the first assertion. Now, using (113) we have

η N p D N p,n (f n ) η N p D N p,n (1)
= Φ p+1 η N p D N p+1,n (f n ) Φ p+1 η N p D N p+1,n (1)
Recalling that D N p,n (1) = Q p,n (1) = G p,n we readily prove (114) and (115). This ends the proof of the lemma.

Additive functional models

In this section we provide a brief discussion on the action of the operators D N p,n and L N p,n on additive linear functionals

f n (x 0 , . . . , x n ) = n p=0 f p (x p ) (118)
associated with some collection of functions f n ∈ Osc(E n ).

D N p,n (f n ) = Q p,n (1)
  0≤q<p M p,η N p-1 . . . M q+1,η N q (f q ) + p≤q≤n R (n) p,q (f q )   RR n°7677
with triangular array of Markov transitions R (n)

p,q introduced in definition 3.1. By definition of L N p,n , we also have that

L N p,n (f n ) = 0≤q<p
M p,η N p-1 . . . M q+1,η N q (f q ) + p≤q≤n R (n) p,q (f q )

using the estimates (53) , we prove the following upper bounds osc(L N p,n (f n ))

≤ 0≤q<p β M p,η N p-1 . . . M q+1,η N q + p≤q≤n g q,n × β (P p,q )

There are many ways to control the Dobrushin operator norm of the product of the random matrices defined in [START_REF] Dawson | Measure-valued Markov processes[END_REF]. For instance, we can use the multiplicative formulae

β M p,η N p-1 . . . M q+1,η N q ≤ p<k≤q β M k+1,η N k
One of the simplest way to proceed, is to assume that

H n (x, y) ≤ τ H n (x, y ′ ) (119)
for any x, y, y ′ , and for some finite constant τ < ∞. In this situation, we find that M k+1,η N k (y, dx) ≤ τ 2 M k+1,η N k (y ′ , dx) from which we conclude that

β M k+1,η N k ≤ 1 -τ -2
We further assume that the condition H m (G, M) stated in section 3.4.1 is met for some m ≥ 1. In this situation, we have osc L N p,n (f n )

≤ 0≤q<p 1τ -2 (p-q) + χ m g m p≤q≤n 1g -(m-1) χ -2 m ⌊(q-p)/m⌋ from which we prove the following uniform estimates sup 0≤p≤n osc L N p,n (f n ) ≤ τ 2 + m g 2m-1 χ 3 m (120)

A stochastic perturbation analysis

As in section 6.3, we develop a stochastic perturbation analysis that allows to express W Γ,N n and W Q,N n in terms of the local sampling random fields (V N p ) 0≤p≤n . These first order expansions presented will be expressed in terms of the first order functions d ν Ψ G (f ) introduced in (99), and the random G N p-1 -measurable functions G N p,n introduced in definition 6.1. Definition 6.6 For any N ≥ 1, any 0 ≤ p ≤ n, and any function f n on the path space E n , we let d N p,n (f n ) be the G N p-1 -measurable functions

d N p,n (f n ) = d Φp(η N p-1 ) Ψ Gp,n L N p,n (f n )
with a second order remain term

R N n (f n ) such that E R N n (f n ) m 1/m ≤ b(2m) 2 r n (f n )
for any m ≥ 1, with some finite constant

r n (f n ) ≤ 4 0≤p≤n g 2 p,n l p,n (f n ) (128)
Proof:

Firstly, we notice that

d N p,n (f n ) ≤ G p,n Φ p (η N p-1 )(G p,n ) osc L N p,n (f n ) ≤ g p,n osc L N p,n (f n )
Using (124), we find the decomposition (127) with the second order remainder term

R N n (f n ) = - n p=0 R N p,n (f n ) with R N p,n (f n ) := 1 η N p (G N p,n ) V N p (G N p,n ) V N p d N p,n (f n )
On the other hand, we have

E R N p,n (f n ) m G N p-1 1/m ≤ g p,n E V N p (G p,n / G p,n ) 2m G N p-1 1/(2m) × E V N p d N p,n (f n ) 2m G N p-1 1/(2m)
Using (101), we prove that

E R N p,n (f n ) m G N p-1 1/m ≤ 4b(2m) 2 g 2 p,n l p,n (f n )
The end of the proof is now clear. This ends the proof of the lemma.

Theorem 6.7 For any N ≥ 1, n ≥ 0, and any bounded function f n on the path space E n , the probability of the events

Q N n -Q n (f n ) ≤ r n (f n ) N 1 + (L ⋆ 0 ) -1 (x) + 2b n σ 2 n (L ⋆ 1 )
-1

x N σ 2 n is greater than 1e -x , for any x ≥ 0, with

σ 2 n := 1 b 2 n 0≤p≤n
g 2 p,n l p,n (f n ) 2 σ 2 p and for any choice of b n ≥ sup 0≤p≤n g p,n l p,n (f n ). In the above displayed formulae, σ n are the uniform local variance parameters defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF], l p,n (f n ) and r n (f n ) are the parameters defined respectively in (126) and (128).

Before getting into the proof of the theorem, we present some direct consequences of these concentration inequalities for normalized additive functional (we use the estimates ( 85) and ( 86)). Corollary 6.8 We assume that the mixing condition H m (G, M) stated in section 3.4.1 is met for some m ≥ 1, and the regularity (119) is satisfied for some finite τ . We also suppose that the parameters σ n defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF] In this notation, for any N ≥ 1, n ≥ 0, and any normalized additive functional f n on the path space E n , the probability of the events

Q N n -Q n (f n ) ≤ c 2 (m) N 1 + (L ⋆ 0 ) -1 (x) + c 1 (m)σ 2 (L ⋆ 1 ) -1
x N (n + 1)σ 2 is greater than 1e -x , for any x ≥ 0. Corollary 6.9 We assume that the assumptions of corollary 6.8 are satisfied, and we set with the constants c 1 (m) and c 2 (m) defined in corollary 6.8. In this situation, the probability of the events

Q N n -Q n (f n ) ≤ 1 N p m,n (x) + 1 √ N q m,n (x)
is greater than 1e -x , for any x ≥ 0.

Proof of theorem 6.7:

We use the same line of arguments as the ones we used in section 6.4.1. Firstly, we notice that On the other hand, we also have that

E V N p δ N p,n (f n ) 2 G N p-1 ≤ 4 (a ⋆ n ) 2 σ 2 p g 2 p,n l p,n (f n ) 2
with a ⋆ := sup 0≤p≤n a p , and the uniform local variance parameters σ 2 p defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF].

This shows that the regularity condition stated in (95) is met by replacing the parameters σ p in the variance formula (95) by the constants 2σ p g p,n l p,n (f n )/a ⋆ n , with the uniform local variance parameters σ p defined in [START_REF] Th | Concentration around the mean for maxima of empirical processes[END_REF].

Using theorem 5.1, we easily prove the desired concentration property. This ends the proof of the theorem.

Empirical processes

Using the same line of arguments as the ones we used in section 6.4.2, we prove the following concentration inequality. Theorem 6.8 We let F n be a separable collection of measurable functions f n on E n , such that f n ≤ 1, osc(f n ) ≤ 1, with finite entropy I(F n ) < ∞. In particular, for any n ≥ 0, and any N ≥ 1, the probability of the following event

sup fn∈Fn Q N n (f n ) -Q n (f n ) ≤ c Fn √ N n p=0
g p,n l p,n Fn x + log 2 is greater than 1e -x , for any x ≥ 0.

Proof: Using (123), for any function f n ∈ Osc(E n ) we have the estimate

W Q,N n (f n ) ≤ 2 n p=0 g p,n l p,n (f n ) V N p (δ N p,n (f n ))
with the G N p-1 -measurable random functions δ N p,n (f n ) on E p defined by

δ N p,n (f n ) = 1 2l p,n (f n ) G p,n G p,n L N p,n (f n ) -Ψ Gp,n Φ p η N p-1 L N p,n (f n )

  get one step further, we denote by f (1) n the empirical mean valued function on E (1)

η 1 M 1 M 1 ⇒

 111 [y0,...,yp-1] p (dx p ) = dp p (x p | (y 0 , . . . , y p-1 ))= dp p (x p | x p-1 ) × p n (x p-1 | (y 0 , . . . , y p-1 )) = η [y0,...,yp-1] p-p (dx p ) and Ψ gp(.,yp) η [y0,...,yp-1] p-p (dx p ) = p(y p |x p ) p p (y p | x ′ p ) dp p (x ′ p | (y 0 , . . . , y p-1 )) dp p (x p | (y 0 , . . . , y p-1 )) = dp p (x p | (y 0 , . . . , y p-1 , y p )) from which we prove that µM p (g p (., y p )) = p p (y p | x p ) dp p (x p | (y 0 , . . . , y p-1 )) = p p (y p | (y 0 , . . . , y p-1 )) and Ψ gp(.,yp) (µM p ) = η [y0,...,yp] p as soon as µ = η [y0,...,yp-1] p-µM p = η [y0,...,yp-1] p

Definition 3 . 1

 31 We denote by Φ p,n (η p ) = η n and γ p Q p,n = γ n with 0 ≤ p ≤ n, the linear and the nonlinear semigroup associated with the unnormalized and the normalized Feynman-Kac measures. For p = n, we use the convention Φ n,n = Id, the identity mapping.

Proposition 3 . 1

 31 For [0, 1]-valued potential function G n , and any couple of measures ν, µ on the set E s.t. µ(G p,n ) ∧ ν(G p,n ) > 0, we have the decomposition

Theorem 4 . 1

 41 For any integer m ≥ 1, and any measurable function f we have the L m -mean error estimates

  for any x ≥ 0. In section 4.3.2 dedicated to concentration properties of random variables Y with finite Orlicz norms π ψ (Y ) < ∞, we shall prove that the probability of the event Y ≤ π ψ (Y ) y + log 2

Proposition 4 . 3

 43 For any random variable B such that

  for any d ≥, any m ≥ 1, and any sequence of real numbers (a i ) 1≤i≤d ∈ R d + , we prove that

Definition 6 . 1

 61 For any 0 ≤ p ≤ n, and any function f on E n , we denote by d p,n (f ) the function on E p defined byd p,n (f ) = d ηp Ψ Gp,n (P p,n (f ))For any N ≥ 1, and any 0 ≤ p ≤ n, we also denote by G N p,n , d N p,n (f ), and d ′N p,n (f ) the G N p-1 -measurable random functions on E p given by

π ψ W η,N n Fn ≤ c Fn τ 1 , 1 1 0log ( 8 +

 n1118 (n) with the parameter τ 1,1 (n) defined in[START_REF] Makrini | Diffusion Monte Carlo method: Numerical analysis in a simple case[END_REF] andc Fn ≤ 24 2 N (F n , ǫ) 2 ) dǫ(106)

  This decomposition allows to reduce the concentration properties of W

W

  any t ∈]0, 1], with g - p := inf x G p (x), we find that R

Corollary 6 . 7

 67 We assume that one of the regularity conditions H m (G, M) stated in section 3.4.1 is met for some m ≥ 0, and we setp m (x) := c 1 (m) 1 + 2(x + √ x) + c 2 (m) x and q m (x) = c 3 (m) √ xwith the parametersc 1 (m) = (4gτ 1,1 (m)) 2 + 8gτ 3,1(m)c 2 (m) = 4(β m g m+1 )/3 and c 3 (m) = 4g 2τ 2,2 (m)σ 2

Definition 6 . 5

 65 Q N n (d(x 0 , . . . , x n )) = η N n (dx n ) Q N n|n (x n , d(x 0 , . . . , x n-1 )) For any 0 ≤ p ≤ n and N ≥ 1, we denote by D N p,n and L N p,n the G N p-1 -measurable integral operators defined byD N p,n (x p , d(y 0 , . . . , y n )) := Q N p|p (x p , d(y 0 , . . . , y p-1 )) δ xp (dy p ) Γ n|p (x p , d(y p+1 , . . . , y n ))(111)andL N p,n (x p , d(y 0 , . . . , y n )) := Q N p|p (x p , d(y 0 , . . . , y p-1 )) δ xp (dy p ) Q n|p (x p , d(y p+1 , . . . , y n ))(112)For p ∈ {0, n}, we use the conventionD N n,n (x n , d(y 0 , . . . , y n )) = L N n,n (x n , d(y 0 , . . . , y n )) = Q N n|n (x n ,d(y 0 , . . . , y n-1 )) δ xn (dy n ) and D N 0,n (x 0 , d(y 0 , . . . , y n )) = δ x0 (dy 0 ) Γ n|0 (x 0 , d(y 1 , . . . , y n )) L N 0,n (x 0 , d(y 0 , . . . , y n )) = δ x0 (dy 0 ) Q n|0 (x 0 , d(y 1 , . . . , y n ))

Lemma 6 . 3

 63 For any 0 ≤ p ≤ n, and any N ≥ 1, and any function f n on the path space E n , we have the almost sure formulaeη N p D N p,n = η N p (G p ) × Φ p+1 (η N p )D N p+1,n

Γ

  n|p (x p , d(y p+1 , . . . , y n )) = Q p+1 (x p , dy p+1 )Γ n|p+1 (x p+1 , d(y p+2 , . . . , y n ))(116) and η N p (dx p )Q p+1 (x p , dy p+1 ) = η N p Q p+1 (dy p+1 ) × M p+1,η N p (y p+1 , dx p )(117)More precisely, we haveη N p (dx p )D N p,n (x p , d(y 0 , . . . , y n )) := η N p (dx p )Q p+1 (x p , dy p+1 )Q N p|p (x p , d(y 0 , . . . , y p-1 )) × δ xp (dy p ) Γ n|p+1 (y p+1 , d(y p+2 , . . . , y n ))Using (117), this implies thatη N p (dx p )D N p,n (x p , d(y 0 , . . . , y n )) := η N p Q p+1 (dy p+1 ) M p+1,η N p (y p+1 , dx p )Q N p|p (x p , d(y 0 , . . . , y p-1 )) × δ xp (dy p ) Γ n|p+1 (y p+1 , d(y p+2 , . . . , y n ))from which we conclude thatη N p D N p,n (f n ) := η N p Q p+1 (dy p+1 ) Q N p+1|p+1 (y p+1 , d(y 0 , . . . , y p ))× Γ n|p+1 (y p+1 , d(y p+2 , . . . , y n )) f n (y 0 , . . . , y n ) = η N p Q p+1 D N p+1,n (f n )

3 m

 3 are uniformly bounded σ = sup n≥0 σ n < ∞ and we set c 1 (m) := 2g m χ m τ 2 + m g 2m-1 χ and c 2 (m) := 2(g m χ m )c 1 (m)

  p m,n (x) = c 2 (m)(1

  d N p,n (f n ) ≤ g p,n l p,n (f n ) δ N p,n (f n ) = d N p,n (f n )/a p ∈ Osc(E p ) ∩ B 1 (E p ) RR n°7677and for any finite constants a p ≥ 2 sup 0≤p≤n g p,n l p,n (f n )

1 0

 1 n l p,n Fn with the functional f n ∈ F n → l p,n (f n ) defined in (126) and c Fn ≤ 24 2 log (8 + N (F n , ǫ) 2 ) dǫ

  1,1 (m)) 2 + 8gτ 3,1 (m)

	and for any p < n
	ς 2 p,n =	4g n	2	(n -p) 2	n -p p≤q<n  1 	g p,q β(P p,q )  	2
	≤	(4g) 2 n	(n -p) n	p≤q<n	g 2 p,q β(P p,q ) 2
	This implies that
	0≤p<n	ς 2 p,n ≤	(4g) 2 n 0≤q<n	τ 2,2 (q) ≤ (4g) 2 τ 2,2

RR n°7677

is a Markov transition and

This implies that

with the Markov transition M δ,µ (x, dy) = 1 2 [M δ (x, dy) + µM δ (x, dy)]

We let {f 1 , . . . , f nǫ } be the n ǫ = N (F , L 2 (ηM δ,µ ) , ǫ/2) centers of L 2 (ηM δ,µ )balls of radius at most ǫ covering F . If we set

then we find that f -

from which we prove that

We conclude that N ((M -µM )(F ), L 2 (η), 2ǫ(1δ)) ≤ N (F , L 2 (ηM δ,µ ) , ǫ)

and therefore N ((M -µM )(F ), 2ǫ(1δ)) ≤ N (F , ǫ)

or equivalently

In more general situations, we quote the following result.

Lemma 5.2 For any ǫ > 0, we have

Proof:

We consider a Hahn-Jordan orthogonal decomposition (M (x, dy) -µM (dy)) = M + µ (x, dy) -M - µ (x, dy) with M + µ (x, dy) = (M (x, .) -µM ) + and M - µ (x, dy) = (M (x, .) -µM ) -

RR n°7677

with M (x, .) -µM tv = M + µ (x, E 1 ) = M - µ (x, E 1 ) ≤ β(M ) By construction, we have

with

and M µ (x, dy) :=

We let {f 1 , . . . , f nǫ } be the n ǫ = N (F , L 2 ηM µ , ǫ/2) centers of L 2 ηM µballs of radius at most ǫ covering F .

If we set

then we find that f -

from which we prove that

In this situation, for any f ∈ (M -µM )(F ), there exists some

We conclude that

and therefore

This ends the proof of the lemma.

we prove that

Using (98), we have

The proof of ( 104) is based on the telescoping sum decomposition

we prove that

with the second order remainder term

Replacing f by the centred function (fη n (f )), and using the fact that

with the second order remainder term

Proof: Using (102), for any function f n ∈ Osc(E n ) we have the estimate

with the G N p-1 -measurable random functions δ N p,n (f n ) on E p defined by

By construction, we have

The end of the proof is now a direct consequence of theorem 5.2. This ends the proof of the theorem.

Corollary 6.3

We consider time homogeneous Feynman-Kac models on some common measurable state space E n = E. We also let F be a separable collection of measurable functions f on E, such that f ≤ 1, osc(f ) ≤ 1, with finite entropy I(F ) < ∞.

We also assume that one of the regularity conditions H m (G, M) stated in section 3.4.1 is met for some m ≥ 0. In this situation, for any n ≥ 0, and any N ≥ 1, the probability of the following event

is greater than 1e -x , for any x ≥ 0.

In the same vein, using the estimates (58), we easily prove the following corollary.

Corollary 6. [START_REF] Bellomo | Modeling in Applied Sciences[END_REF] We also assume that one of the regularity conditions H m (G, M) stated in section 3.4.1 is met for some m ≥ 0.

We let F n be a separable collection of measurable functions f n on the path space E n , such that

We also let η N n be the occupation measure of the genealogical tree model presented in [START_REF] Johansen | Sequential Monte Carlo Samplers for Rare Events[END_REF]. In this situation, for any n ≥ 0, and any N ≥ 1, the probability of the following event

is greater than 1e -x , for any x ≥ 0 with the constant c Fn defined in (106).

RR n°7677

The following corollaries are a direct consequence of (75).

Corollary 6.5 We assume that the conditions stated in corollary 6.3 are satisfied. When F stands for the indicator functions (74) of cells in E = R d , for some d ≥ 1, the probability of the following event

is greater than 1e -x , for any x ≥ 0, for some universal constant c < ∞ that doesn't depend on the dimension.

Corollary 6.6 We assume that the conditions stated in corollary 6.4 are satisfied. When F n stands for product functions of indicator of cells ( 74) in the path space E n = R d0 × . . . , ×R dn , for some d p ≥ 1, p ≥ 0, the probability of the following event

is greater than 1e -x , for any x ≥ 0, for some universal constant c < ∞ that doesn't depend on the dimension.

6.5 Particle free energy models

introduction

The main aim of this section is to analyze the concentration properties of the particle free energy models introduced in section 1.4.2. More formally, the unnormalized particle random field models discussed in this section are defined below.

Definition 6.2 We denote by γ N n the normalized models defined by the following formulae

with the normalized potential functions

We also let W γ,N n and W γ,N n be the random field particle models defined by

These unnormalized particle models γ N n have a particularly simple form. They are defined in terms of product of empirical mean values η N p (G p ) of the potential functions G p w.r.t. the flow of normalized particle measures η N p after the p-th mutation stages, with p < n.

Thus, the concentration properties of γ N n should be related in some way to the one of the interacting processes η N n developed in section 6.4.

Proof:

We use the telescoping sum decomposition

The end of the proof of the first decomposition is now easily completed. We prove (108) using the following formulae

The proof of ( 109) is based on the fact that

for any positive numbers x, y. Indeed, we have the formula

This ends the proof of the theorem.

Concentration inequalities

Combining the exponential formulae (109) with the expansions (108), we derive first order decompositions for the random sequence √ N log γ N n (1). These expansions will be expressed in terms of the random predictable functions defined below.

Definition 6.3

We let h N q,n be the random G N q-1 -measurable functions given by

with the functions d N q,p (G p ) given in definition 6.1.

Lemma 6.2 For any n ≥ 0 and any N ≥ 1, we have

with a second remainder order term R N n such that

for any m ≥ 1, with some constant

Proof:

Using the exponential formulae (109), we have

This implies that

with the (negative) second order remainder term

On the other hand, using (108) we have

with the second order remainder order term

This gives the decomposition (110), with the second remainder order term

Using the fact that

This ends the proof of the lemma.

We are now in position to state and to prove the following concentration theorem. Theorem 6.5 For any N ≥ 1, ǫ ∈ {+1, -1}, n ≥ 0, and for any ς ⋆ n ≥ sup 0≤q≤n ς q,n with ς q,n := 4 n q≤p<n g q,p g p β(P q,p ) the probability of the following events

is greater than 1e -x , for any x ≥ 0, with the parameters

Before getting into the proof of the theorem, we present simple arguments to derive exponential concentration inequalities for the quantities γ N n (1) -1 . Suppose that for any ǫ ∈ {+1, -1}, the probability of events

Definition 6.4 We let W Γ,N n and W Q,N n be random field models defined by

The analysis of the fluctuation random fields of backward particle models is a little more involved than the one of the genealogical tree particle models. The main difficulty is to deal with the nonlinear dependency of these backward particle Markov chain models with the flow of particle measures η N n . In section 6.6.1, we provide some preliminary key backward conditioning principles. We also introduce some predictable integral operators involved in the first order expansions of the fluctuation random fields discussed in section 6.6.3. In section 6.6.2, we illustrate these models in the context of additive functional models. In section 6.6.4, we put together the semigroup techniques developed in earlier sections to derive a series of quantitative concentration inequalities.

Some preliminary conditioning principles

By definition of the unnormalized Feynman-Kac measures Γ n , we have

This implies that

with the Q n -distribution of the random states (X 0 , . . . , X p )

and the Q n -conditional distribution of (X p+1 , . . . , X n ) given the random state X p = x p defined by

Now, we discuss some backward conditioning principles. Using the backward Markov chain formulation (8), we have

with the Q n -conditional distribution of (X 0 , . . . , X n-1 ) given the terminal random state X n = x n defined by the backward Markov transition

By construction, the Q N n -conditional distribution of (X 0 , . . . , X n-1 ) given the terminal random state X n = x n is also defined by the particle backward Markov transition given by

M q,η N q-1 (x q , dx q-1 )

RR n°7677

We are now in position to state and to prove the following decomposition theorem. Theorem 6.6 For any 0 ≤ p ≤ n, and any function f n on the path space E n+1 , we have

In addition, we have

To prove the first assertion, we use a backward induction on the parameter p. For p = n, the result is immediate since we have

We suppose that the formula is valid at a given rank p ≤ n. In this situation, using the fact that

Applying (113), we also have that

from which we conclude that the desired formula is satisfied at rank (p -1). This ends the proof of first assertion. Now, combining lemma 6.3 and (121), the proof of the second assertion is simply based on the following decomposition

To prove the final decomposition, we use the fact that

with the conventions η N -1 D N -1,n = η 0 Γ n|0 , for p = 0. Finally, we use ( 114) and (115) to check that

We end the proof using the first order expansions of Boltzmann-Gibbs transformation developed in section 98

This ends the proof of the theorem.

Concentration inequalities

Finite marginal models Given a bounded function f n on the path space E n , we further assume that we have some almost sure estimate

for some finite constant l p,n (f n ) (≤ f n ). For instance, for additive functionals of the form (118), we have proved in section 6.6.2 the following uniform estimates

which are valid for any N ≥ 1 and any 0 ≤ p ≤ n; as soon as the mixing condition H m (G, M) stated in section 3.4.1 is met for some m ≥ 1, and the regularity (119) is satisfied for some finite τ . For any additive functional f n of the form (118), we denote by f n = f n /(n+1) the normalized additive functional. Lemma 6.4 For any N ≥ 1, n ≥ 0, and any bounded function f n on the path space E n , we have the first order decomposition

By construction, we have

Using the uniform estimate (100), if we set

The end of the proof is now a direct consequence of theorem 5.2, and theorem 4.2. This ends the proof of the theorem. Definition 6.7 We let F = (F n ) n≥0 , be a sequence of separable collections F n of measurable functions f n on E n , such that f n ≤ 1, osc(f n ) ≤ 1, and finite entropy I(F n ) < ∞.

For any n ≥ 0, we set

We also denote by Σ n (F ) the collection of additive functionnals defined by

Theorem 6.9 For any N ≥ 1, and any n ≥ 0, we have

with some constant

and for any a collection of [0, 1]-valued parameters β p,q such that

Proof: By definition of the operator d N p,n given in definition 6.6, for additive functionals f n of the form (118), we find that

This implies that

By construction, we have that δ (N,i) p,q,n (f q ) ≤ 1/2 and osc δ (N,i) p,q,n (f q ) ≤ 1

for any i ∈ {1, 2}. We set

Using the uniform estimate (100), we also prove the almost sure upper bound sup N ≥1 N G (N,i) p,q,n , ǫ ≤ N (F q , ǫ/2) Using theorem 5.2, we prove that

The end of the proof of the theorem is now a direct consequence of the decomposition (123). This ends the proof of the theorem.

Corollary 6.10 We further assume that the condition H m (G, M) stated in section 3.4.1 is met for some m ≥ 1, the condition (119) is satisfied for some τ , and we have J(F ) := sup n≥0 J n (F ) < ∞. In this situation, we have the uniform estimates

with some constant c F (m) ≤ χ m g m τ 2 + m g 2m-1 χ 3 m J(F )

In particular, for any time horizon ≥ 0, and any N ≥ 1, the probability of the following event

is greater than 1e -x , for any x ≥ 0.

Proof:

When the conditions H m (G, M) and ( 119) are satisfied, we proved in section 6.6.2 that 0≤q<p β p,q + p≤q≤n β R (n) p,q ≤ τ 2 + m g 2m-1 χ 3 m

We end the proof of the theorem, recalling that g p,n ≤ χ m g m . This ends the proof of the theorem.

We end this section with a direct consequence of (75).

Corollary 6.11 We further assume that the condition H m (G, M) stated in section 3.4.1 is met for some m ≥ 1, the condition (119) is satisfied for some τ . We let F n be the set of product functions of indicator of cells in the path space E n = R d , for some d ≥ 1, p ≥ 0. In this situation, for any time horizon n ≥ 0, and any N ≥ 1, the probability of the following event

is greater than 1e -x , for any x ≥ 0, with some constant c(m) ≤ c χ m g m τ 2 + m g 2m-1 χ 3 m

In the above display, c stands for some finite universal constant.