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Processus de Levy: R.A. Doney: Tanaka's construction for random walks and Levy processes.- R.A. Doney: Some excursion calculations for spectrally one-sided Levy processes.- A.E. Kyprianou, Z. Palmowski: A martingale review of some fluctuation theory for spectrally megative Levy processes.- M.R. Pistorius: A potential-theoretical review of some exit problems of spectrally negative Levy processes.- L. Nguyen-Ngoc, M. Yor: Some martingales associated to reflected Levy processes.- K.B. Erickson, R.A. Maller: Generalised Ornstein-Uhlenbeck processes and the convergence of Levy integrals.- Autres Exposes: P. Fougeres: Spectral gap for log-concave probability measures on the real line.- L. Godefroy: Propriete de Choquet-Deny et fonctions harmoniques sur les hypergroupes commutatifs.- M. Buiculescu: Exponential decay parameters associated with excessive measures.- V. Grecca: Positive bilinear mappings associated with stochastic processes.- A. Jakubowski: An almost sure approximation for the predictable process in the Doob-Meyer decomposition theorem.- A. Cherny, A. Shiryaev: On stochastic integrals up to infinity and predictable criteria for integrability.- Y. Kabanov, C. Stricker: Remarks on the true no-arbitrage property.- H. Buhler: Information-equivalence: On filtrations created by independent increments.- M. Zakai: Rotations and tangent processes on Wiener space.- I. Shigekawa: Lp multiplier theorem for the Hodge-Kodaira operator.- G. Peccati, C.A. Tudor: Gaussian limits for vector-valued multiple stochastic integrals.- J. Rosen: Derivatives of self-intersection local times.- N. Eisenbaum, C. A. Tudor: On squared fractional Brownian motions.- A. Ayache et al: Regularity and identification of generalised miltifractional Gaussian processes.- F. Benaych-Georges: Failure of the Raikov theorem for free random variables.- G. Aubrun: Aharp small deviation inequality for the largest eigenvalue of a randommatrix.- F. Baudoin: The tangent space to a hypoelliptic diffusion and applications.- A. Bencherif-Madani, E. Pardoux: Homogenization of a diffusion with locally periodic coefficients.

We examine the Charlier, Meixner, Krawtchouk and Hahn discrete orthogonal polynomial ensembles, deeply investigated by K. Johansson, using integration by parts for the underlying Markov operators, differential equations on Laplace transforms and moment equations. As for the matrix ensembles, equilibrium measures are described as limits of empirical spectral distributions. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. Factorial moment identities on mean spectral measures may be used towards small deviation inequalities on the rightmost charges at the rate given by the Tracy-Widom asymptotics.

Following the investigation by U. Haagerup and S. Thorbjornsen, we present a simple differential approach to the limit theorems for empirical spectral distributions of complex random matrices from the Gaussian, Laguerre and Jacobi Unitary Ensembles. In the framework of abstract Markov diffusion operators, we derive by the integration by parts formula differential equations for Laplace transforms and recurrence equations for moments of eigenfunction measures. In particular, a new description of the equilibrium measures as adapted mixtures of the universal arcsine law with an independent uniform distribution is emphasized. The moment recurrence relations are used to describe sharp, non asymptotic, small deviation inequalities on the largest eigenvalues at the rate given by the Tracy-Widom asymptotics.

Cours specialises et exposes thematiques: A. Guionnet, B. Zegarlinski: Lectures on Logarithmic Sobolev inequalities.- A. Lejay, L. Pastur: Matrices aleatoires : statistique asymptotique des valeurs propres.- N. O'Connell: Random matrices, non-colliding particle systems and queues.- Exposes: A. Dermoune, O. Moutsinga: Generalized variational principles.- D. Chafai: Gaussian maximum of entropy and reversed log-Sobolev inequality.- L. Miclo: About projections of logarithmic Sobolev inequalities.- L. Miclo: Sur l'inegalite de Sobolev logarithmique des operateurs de Laguerre a petit parametre.- A. Bentaleb: Sur les fonctions extremales des inegalites de Sobolev des operateurs de diffusion.- C. Donati-Martin, Y. Hu: Penalization of the Wiener measure and principal values.- C. Leuridan: Theoreme de Ray-Knight dans un arbre : Une approche algebrique.- R. Bass: Stochastic differential equations driven by symmetric stable processes.- T. Simon: Support d'une equation d' Ito avec sauts en dimension 1.- N. Eisenbaum: A Gaussian sheet connected to symmetric Markov chains.- C. Leuridan: Filtration d'une marche aleatoire stationnaire sur le cercle.- S. Beghdadi-Sakrani: Une martingale non pure, dont la filtration est brownienne.- J. Hannig: On filtrations related to purely discontinuous martingales.- S. Beghdadi-Sakrani: Calcul stochastique pour des mesures signees.- J. Jacod: On processes with conditional independent increments and stable convergence in law.- V. Grecea: Duality and quasi-continuity for supermartingales.- Y. Kabanov, C. Stricker: On the true submartingale property, d'apres Schachermayer.- C. Stricker: Simple strategies in exponential utility maximization.- M. Arnaudon, A. Thalmaier: Horizontal martingales in vector bundles.- D. Kurtz: Representation nucleaire des martingales d'Azema.- S. Attal: Approximating the Fock space with the toy Fock space.- Corrections auxvolumes anterieurs.

Preface.- F.B. Knight: An Impression of P.A. Meyer as Deus Ex Machina.- Advanced course: A. Lejay: An introduction to rough paths.- Talks: D. Bakry, O. Mazet : Characterization of Markov semigroups on R associated to some families of orthogonal polynomials.- P. Cheridito: Representations of Gaussian measures that are equivalent to Wiener measure.- L. Galtchouk: On the reduction of a multidimensional continuous martingale to a Brownian motion.- I. Meilijson: The time to a given drawdown in Brownian motion.- A. Lachal: Application de la theorie des excursions a l'integrale du mouvement Brownien.- T. Mountford: Brownian sheet local time and bubbles.- K. Hirano: On the maximum of a diffusion process in a random Levy environment.- D. Khoshnevisan: The codimension of the zeros of a stable process in random scenery.- J. Brossard: Deux notions equivalentes d'unicite en loi pour les equations differentielles stochastiques.- Z. Brzezniak, A. Carroll : Approximation of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces.- F. Delarue: Estimates of the solutions of a system of quasi-linear PDEs. A probabilistic scheme.- G. Miermont, J. Schweinsberg: Self-similar fragmentations and stable subordinators.- M. Ledoux: A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices.- Ya. Doumerc: A note on representations of eigenvalues of classical Gaussian matrices.- E. Strasser: Necessary and sufficient conditions for the supermartingale property of a stochastic integral with respect to a local martingale.- M. Rasonyi: A remark on the superhedging theorem under transaction costs.- I. Rosu, D. Stroock: On the derivation of the Black-Scholes formula.- P. del Moral, A. Doucet: On a class of genealogical and interacting Metropolis models.

We point out a simple argument relying on hypercontractivity to describe tail inequalities on the distribution of the largest eigenvalues of random matrices at the rate given by the Tracy–Widom distribution. The result is illustrated on the known examples of the Gaussian and Laguerre unitary ensembles. The argument may be applied to describe the generic tail behavior of eigenfunction measures of hypercontractive operators.

This research project is related to the control of combustion instability occurring intermittently in aeroengines. The unsteadiness of the flow is induced by a particular instability called Combustion-Driven Oscillations (CDO).

We analyze recent proofs of decay of correlations and logarithmic Sobolev inequalities for unbounded spin systems in the perturbative regime developed by B. Zegarlinski, N. Yoshida, B. Helffer, Th. Bodineau. We investigate to this task a simple analytic model. Proofs are short and self-contained.

International audience; Some Gaussian functional inequalities have simple generalizations to some Gaussianlike cases. They allow us to establish Gaussian-like Large Deviations Principles and bounds via Gaussian concentration and shift inequalities for certain families of Boltzmann measures and laws of diffusion semigroups in short time. Beyond the results themselves, we would like to emphasize here the method and the symmetry of the arguments used for upper and lower bounds by means of the functional inequalities.

Sous-mesures symetriques sur un ensemble fini.- Sur une inegalite de Sobolev logarithmique pour une diffusion unidimensionnelle.- Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les operateurs de Jacobi et de Laguerre.- Quand l'inegalite log-Sobolev implique l'inegalite de trou spectral.- Trous spectraux a basse temperature: un contre-exemple a un comportement asymptotique escompte.- Some remarks on the optional decomposition theorem.- Separation d'une sur-et d'une sousmartingale par une martingale.- Closedness of some spaces of stochastic integrals.- Homogeneous diffusions on the Sierpinski gasket.- Almost sure path properties of Branching Diffusion Processes.- Criteria of regularity at the end of a tree.- Normalized stochastic integrals in topological vector spaces.- Pathwise uniqueness and approximation of solutions of stochastic differential equations.- Stability of stochastic differential equations in manifolds.- Propagation trajectorielle du chaos pour les lois de conservation scalaire.- Some calculations for perturbed Brownian motion.- Perturbed bessel processes.- The maximum maximum of a martingale.- Autour d'un theoreme de tsirelson sur des filtrations Browniennes et non Browniennes.- Sur un theoreme de tsirelson relatif a des mouvements browniens correles et a la nullite de certains temps locaux.- A remark on Slutsky's theorem.- Quelques calculs de compensateurs impliquant l'injectivite de certains processus croissants.- The Brownian Burglar: conditioning Brownian motion by its local time process.- On the upcrossing chains of stopped Brownian motion.- Le theoreme de ray-knight a temps fixe.- Point le plus visite par un mouvement brownien avec derive.- Brownian motion, excursions, and matrix factors.- Sur les temps de coupure des marches aleatoires reflechies.

We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition.

We present short and elementary proofs of three recent results on exponential integrability of Lipschitz functions and quantitative bounds on the diameter under logarithmic Sobolev inequalities due respectively to S. Aida, T. Masuda, I. Shigekawa [A-M-S], D. Bakry, D. Michel [B-M] and L. Saloff-Coste [SC]. Although the first two results we aim to simplify deal with abstract Markov generators on probability spaces, we would like to briefly present the purpose of this note in the setting of the Laplace-Beltrami operator ∆ on a complete connected Riemannian manifold M of finite volume V . We will consider the normalized measure dμ = 1 V dv where dv denote the Riemannian measure and let ∇ be the Riemannian gradient on M . For a nonnegative bounded (say) real valued function f on M , let E(f) denote the entropy of f with respect to μ defined by E(f) = ∫ f log fdμ− ∫ fdμ log (∫ fdμ ) . We will say that ∆ satisfies a logarithmic Sobolev inequality if there exists ρ > 0 such that for all C∞, compactly supported or bounded, functions f on M , ρE(f) ≤ 2 ∫ f(−∆f)dμ = 2 ∫ |∇f |dμ. The largest possible value ρ0 for ρ is called the logarithmic Sobolev constant of the Laplacian ∆ on M , or simply of M . More generaly, one may consider, following [B], inequalities between entropy and energy of the type E(f) ≤ Φ (∥∥|∇f |∥∥2 2 ) for all C∞ bounded functions f with ‖f‖2 = 1 where Φ is a nonnegative function on [0,∞). With these notations, S. Aida, T. Masuda and I. Shigekawa [A-M-S] recently showed that, when ρ0 > 0, whenever f is a function on M such that ‖|∇f |‖∞ ≤ 1 (that is f is Lipschitz with Lipschitz norm less than or equal to 1), then, for every 0 < α < ρ0/2

The study of strong limit theorems for sums of independent random variables such as the strong law of large numbers or the law of the iterated logarithm in the preceding chapters showed that in Banach spaces these can only be reasonably understood when the corresponding weak property, that is tightness or convergence in probability, is satisfied. It was shown indeed that under some natural moment conditions, the strong statements actually reduce to the corresponding weak ones. On the line, or in finite dimensional spaces, the moment conditions usually automatically ensure the weak limiting property. As we pointed out, this is no longer the case in general Banach spaces.

In this chapter and in the next one, we present respectively the strong law of large numbers and the law of the iterated logarithm for sums of independent Banach space valued random variables. In this study, the isoperimetric approach of Section 6.3 demonstrates its efficiency. We only investigate extensions to vector valued random variables of some of the classical limit theorems such as the laws of large numbers of Kolmogorov and Prokhorov.

Sums of independent random variables already appeared in the preceding chapters in some concrete situations (Gaussian and Rademacher averages, representation of stable random variables). On the intuitive basis of central limit theorems which approximate normalized sums of independent random variables by smooth limiting distributions (Gaussian, stable), one would expect that results similar to those presented previously should hold in a sense or in another for sums of independent random variables. The results presented in this chapter go in this direction and the reader will recognize in this general setting the topics covered before: integrability properties, equivalence of moments, concentration, tail behavior, etc. We will mainly describe ideas and techniques which go from simple but powerful observations such as symmetrization (randomization) techniques to more elaborate results like those obtained from the isoperimetric inequality for product measures of Theorem 1.4. Section 6.1 is concerned with symmetrization, Section 6.2 with Hoffmann-Jorgensen’s inequalities and the equivalence of moments of sums of independent random variables. In the last and main section, martingale and isoperimetric methods are developed in this context. Many results presented in this chapter will be of basic use in the study of limit theorems later.

The notion of type of a Banach space already appeared in the last chapters on the law of large numbers and the law of the iterated logarithm. We observed there that, in quite general situations, almost sure properties can be reduced to properties in probability or in L P , 0 ≤ p < ∞. Starting with this chapter, we will now study the possibility of a control in probability, or in the weak topology, of probability distributions of sums of independent random variables.

In Chapter 11, we evaluated random processes indexed by an arbitrary index set T. In this chapter, we take advantage of some homogeneity properties of T and we investigate in this setting, using the general conclusions of Chapters 11 and 12, the more concrete random Fourier series. The tools developed so far indeed lead to a definitive treatment of those processes with applications to Harmonic Analysis. Our main reference for this chapter is the work by M. B. Marcus and G. Pisier [M-P1], [M-P2] to which we refer for an historical background and accurate references and priorities.

In Chapter 9 we described how certain conditions on Banach spaces can ensure the existence and the tightness of some probability measures.

This chapter is devoted to the classical laws of the iterated logarithm of Kolmogorov and Hartman-Wintner-Strassen in the vector valued setting. These extensions both enlighten the scalar statements and describe various new interesting phenomena in the infinite dimensional setting. As in the previous chapter on the strong law of large numbers, the isoperimetric approach proves to be an efficient tool in this study. The main results described here show again how the strong almost sure statement of the law of the iterated logarithm reduces to the corresponding (necessary) statement in probability, under moment conditions similar to those of the scalar case.

In this first chapter, we present the isoperimetric inequalities which now appear as the crucial concept in the understanding of various concentration inequalities, tail behaviors and integrability theorems in Probability in Banach spaces. These inequalities often arise as the final and most elaborate forms of previous, weaker (but already efficient) inequalities which will be mentioned in their framework throughout the book. In these final forms however, the isoperimetric inequalities and associated concentration of measure phenomena provide the appropriate ideas for an in depth comprehension of some of the most important theorems of the theory.

This chapter collects in rather an informal way some basic facts about processes and infinite dimensional random variables. The material that we present actually only appears as the necessary background for the subsequent analysis developed in the next chapters. Only a few proofs are given and many important results are only just mentioned or even omitted. It is therefore recommended to complement, if necessary, these partial bases with the classical references, some of which are given at the end of the chapter.

The purpose of this chapter is to present applications of the random process techniques developed so far to infinite dimensional limit theorems, and in particular to the central limit theorem (CLT). More precisely, we will be interested for example in the CLT in the space C(T) of continuous functions on a compact metric space T. Since C(T) is not well behaved with respect to the type or cotype 2 properties, we will rather have to seek for nice classes of random variables in C(T) for which a central limit property can be established. This point of view leads to enlarge this framework and to investigate limit theorems for empirical measures or processes. Random geometric descriptions of the CLT may then be produced via this approach, as well as complete descriptions for nice classes of functions (indicator functions of some sets) on which the empirical processes are indexed. While these random geometric descriptions do not solve the central limit problem in infinite dimension (and are probably of little use in applications), however, they clearly describe the main difficulties inherent to the problem from the empirical point of view.

In the preceding chapter, we presented some sufficient metric entropy and majorizing measure conditions for the sample boundedness and continuity of random processes satisfying incremental conditions. In particular, these results were applied to Gaussian random processes in Section 11.3. The main concern of this chapter is necessity. We will see indeed, as one of the main results, that the sufficient majorizing measure condition for a Gaussian process to be almost surely bounded or continuous is actually also necessary. This characterization thus provides a complete understanding of the regularity properties of Gaussian paths. The arguments of proof rely heavily on the basic ultrametric structure which lies behind a majorizing measure condition.

With this chapter, we really enter into the subject of Probability in Banach spaces. The study of Gaussian random vectors and processes may indeed be considered as one of the fundamental topics of the theory since it inspires many other parts of the field both in the results themselves and in the techniques of investigation. Historically, the developments also followed this line of progress.

This paper describes a method for the quantitative analysis of solutions containing a mixture of proteins and sulfobetaine. In a preliminary step the proteins, which interfere with the detergent assay, are separated by precipitation with trichloroacetic acid (8%). The insoluble fraction, dissolved in NaOH (1.0 N), and the soluble fraction, containing the detergent, are treated with the Folin-Ciocalteu phenol reagent, essentially following the method of O. H. Lowry, N. J. Rosebrough, A. L. Farr, and R. J. Randall (1951, J. Biol. Chem. 193, 265-275). The absorbance of the protein fraction is read, as usual at 750 nm, while that of the detergent solution is read at 342 nm. At this wavelength, sulfobetaine, treated with the Folin reagent, absorbs strongly, the absorbances being proportional to its concentration up to 1.5 mg/ml.

We give necessary and sufficient conditions for a random variable X X with values in a uniformly convex Banach space B B to satisfy the law of the iterated logarithm. Precisely, we show that a B B -valued random variable X X satisfies the (compact) law of the iterated logarithm if and only if E { | | X | | 2 / L 2 | | X | | } > ∞ E\{ ||X|{|^2}/{L_2}||X||\} > \infty , the family { | x ∗ ( X ) | 2 ; x ∗ ∈ B ∗ , | | x ∗ | | = 1 } \{ |{x^{\ast }}(X){|^2};\,{x^{\ast }} \in {B^{\ast }},\,||{x^{\ast }}|| = 1\} is uniformly integrable and S n / a n → 0 {S_n}/{a_n} \to 0 in probability.