In Memoriam Marc Yor - Séminaire de Probabilités XLVII

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Integral Representations of Certain Measures in the One-Dimensional Diffusions Excursion Theory

Abstract

In this note we present integral representations of the Itô excursion measure associated with a general one-dimensional diffusion X. These representations and identities are natural extensions of the classical ones for reflected Brownian motion, RBM. As is well known, the three-dimensional Bessel process, BES(3), plays a crucial rôle in the analysis of the Brownian excursions. Our main interest is in showing explicitly how certain excursion theoretical formulae associated with the pair (RBM, BES(3)) generalize to pair (X, X ^↑ ), where X ^↑ denotes the diffusion obtained from X by conditioning X not to hit 0. We illustrate the results for the pair \((R_{-},R_{+})\) consisting of a recurrent Bessel process with dimension \(d_{-} = 2(1-\alpha ),\) α ∈ (0, 1), and a transient Bessel process with dimension \(d_{+} = 2(1+\alpha )\). Pair (RBM, BES(3)) is, clearly, obtained by choosing \(\alpha = 1/2.\)

Paavo Salminen, Ju-Yi Yen, Marc Yor

Sticky Particles and Stochastic Flows

Abstract

Gawȩdzki and Horvai have studied a model for the motion of particles carried in a turbulent fluid and shown that in a limiting regime with low levels of viscosity and molecular diffusivity, pairs of particles exhibit the phenomena of stickiness when they meet. In this paper we characterise the motion of an arbitrary number of particles in a simplified version of their model.

Jon Warren

Infinitesimal Invariance for the Coupled KPZ Equations

Abstract

This paper studies the infinitesimal invariance for \(\mathbb{R}^{d}\)-valued extension of the Kardar-Parisi-Zhang (KPZ) equation at approximating level.

Tadahisa Funaki

Patterns in Random Walks and Brownian Motion

Abstract

We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.

Jim Pitman, Wenpin Tang

Bessel Processes, the Brownian Snake and Super-Brownian Motion

Abstract

We prove that, both for the Brownian snake and for super-Brownian motion in dimension one, the historical path corresponding to the minimal spatial position is a Bessel process of dimension − 5. We also discuss a spine decomposition for the Brownian snake conditioned on the minimizing path.

Jean-François Le Gall

On Inversions and Doob h-Transforms of Linear Diffusions

Abstract

Let X be a regular linear diffusion whose state space is an open interval \(E \subseteq \mathbb{R}\). We consider the dual diffusion X ^∗ whose probability law is obtained as a Doob h-transform of the law of X, where h is a positive harmonic function for the infinitesimal generator of X on E. We provide a construction of X ^∗ as a deterministic inversion I(X) of X, time changed with some random clock. Such inversions generalize the Euclidean inversions that intervene when X is a Brownian motion. The important case where X ^∗ is X conditioned to stay above some fixed level is included. The families of deterministic inversions are given explicitly for the Brownian motion with drift, Bessel processes and the three-dimensional hyperbolic Bessel process.

Larbi Alili, Piotr Graczyk, Tomasz Żak

On h-Transforms of One-Dimensional Diffusions Stopped upon Hitting Zero

Abstract

For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are h-transforms of the process stopped upon hitting zero, where h’s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the h-transforms are investigated.

Kouji Yano, Yuko Yano

h-Transforms and Orthogonal Polynomials

Abstract

We describe some examples of classical and explicit h-transforms as particular cases of a general mechanism, which is related to the existence of symmetric diffusion operators having orthogonal polynomials as spectral decomposition.

Dominique Bakry, Olfa Zribi

On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration

Abstract

Given a reference filtration \(\mathbb{F}\), we consider the cases where an enlarged filtration \(\mathbb{G}\) is constructed from \(\mathbb{F}\) in two different ways: progressively with a random time or initially with a random variable. In both situations, under suitable conditions, we present a \(\mathbb{G}\)-optional semimartingale decomposition for \(\mathbb{F}\)-local martingales. Our study is then applied to the question of how an arbitrage-free semimartingale model is affected when stopped at the random time in the case of progressive enlargement or when the random variable used for initial enlargement satisfies Jacod’s hypothesis. More precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition, also called non arbitrages of the first kind in the literature. We provide alternative proofs of some results from Aksamit et al. (Non-arbitrage up to random horizon for semimartingale models, short version, preprint, 2014 [arXiv:1310.1142]), incorporating a different methodology based on our optional semimartingale decomposition.

Anna Aksamit, Tahir Choulli, Monique Jeanblanc

Martingale Marginals Do Not Always Determine Convergence

Abstract

Baéz-Duarte (J. Math. Anal. Appl. 36, 149–150, 1971, http://​dx.​doi.​org/​10.​1016/​0022-247X(71)90025-4 [ISSN 0022-247x]) and Gilat (Ann. Math. Stat. 43, 1374–1379, 1972, http://​dx.​doi.​org/​10.​1214/​aoms/​1177692494 [ISSN 0003-4851]) gave examples of martingales that converge in probability (and hence in distribution) but not almost surely. Here such a martingale is constructed with uniformly bounded increments, and a construction is provided of two martingales with the same marginals, one of which converges almost surely, while the other does not converge in probability.

Jim Pitman

Martingale Inequalities for the Maximum via Pathwise Arguments

Abstract

We study a class of martingale inequalities involving the running maximum process. They are derived from pathwise inequalities introduced by Henry-Labordère et al. (Ann. Appl. Probab., 2015 [arxiv:1203.6877v3]) and provide an upper bound on the expectation of a function of the running maximum in terms of marginal distributions at n intermediate time points. The class of inequalities is rich and we show that in general no inequality is uniformly sharp—for any two inequalities we specify martingales such that one or the other inequality is sharper. We use our inequalities to recover Doob’s L ^p inequalities. Further, for p = 1 we refine the known inequality and for p < 1 we obtain new inequalities.

Jan Obłój, Peter Spoida, Nizar Touzi

Polynomials Associated with Finite Markov Chains

Abstract

Given a finite Markov chain, we investigate the first minors of the transition matrix of a lifting of this Markov chain to covering trees. In a simple case we exhibit a nice factorisation of these minors, and we conjecture that it holds more generally.

Philippe Biane

On σ-Finite Measures Related to the Martin Boundary of Recurrent Markov Chains

Abstract

In our monograph with Roynette and Yor (Najnudel et al., A Global View of Brownian Penalisations, MSJ Memoirs, vol. 19, Mathematical Society of Japan, Tokyo, 2009), we construct a σ-finite measure related to penalisations of different stochastic processes, including the Brownian motion in dimension 1 or 2, and a large class of linear diffusions. In the last chapter of the monograph, we define similar measures from recurrent Markov chains satisfying some technical conditions. In the present paper, we give a classification of these measures, in function of the minimal Martin boundary of the Markov chain considered at the beginning. We apply this classification to the examples considered at the end of Najnudel et al. (A Global View of Brownian Penalisations, MSJ Memoirs, vol. 19, Mathematical Society of Japan, Tokyo, 2009).

Joseph Najnudel

Loop Measures Without Transition Probabilities

Abstract

The goal of this paper is to define and study loop measures for Markov processes without transition densities. In particular, we prove the shift invariance of the based loop measure.

Pat Fitzsimmons, Yves Le Jan, Jay Rosen

The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk

Abstract

A complete characterization of the possible joint distributions of the maximum and terminal value of uniformly integrable martingale has been known for some time, and the aim of this paper is to establish a similar characterization for continuous martingales of the joint law of the minimum, final value, and maximum, along with the direction of the final excursion. We solve this problem completely for the discrete analogue, that of a simple symmetric random walk stopped at some almost-surely finite stopping time. This characterization leads to robust hedging strategies for derivatives whose value depends on the maximum, minimum and final values of the underlying asset.

Moritz Duembgen, L. C. G. Rogers

Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach

Abstract

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators \(\Gamma _{i}\) (introduced by Nourdin and Peccati, J. Appl. Funct. Anal. 258(11), 3775–3791, 2010), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly (Electron. Commun. Probab. 17(36), 1–12, 2012), concerning the limiting behavior of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

Ehsan Azmoodeh, Giovanni Peccati, Guillaume Poly

Mod-Gaussian Convergence and Its Applications for Models of Statistical Mechanics

Abstract

In this paper we complete our understanding of the role played by the limiting (or residue) function in the context of mod-Gaussian convergence. The question about the probabilistic interpretation of such functions was initially raised by Marc Yor. After recalling our recent result which interprets the limiting function as a measure of “breaking of symmetry” in the Gaussian approximation in the framework of general central limit theorems type results, we introduce the framework of L¹-mod-Gaussian convergence in which the residue function is obtained as (up to a normalizing factor) the probability density of some sequences of random variables converging in law after a change of probability measure. In particular we recover some celebrated results due to Ellis and Newman on the convergence in law of dependent random variables arising in statistical mechanics. We complete our results by giving an alternative approach to the Stein method to obtain the rate of convergence in the Ellis-Newman convergence theorem and by proving a new local limit theorem. More generally we illustrate our results with simple models from statistical mechanics.

Pierre-Loïc Méliot, Ashkan Nikeghbali

On Sharp Large Deviations for the Bridge of a General Diffusion

Abstract

We provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit probability are independent of the drift b, provided it satisfies a simple condition that is always satisfied in dimension 1. On the other hand we produce an example where this assumption is not satisfied and the drift is actually influential.

Paolo Baldi, Lucia Caramellino, Maurizia Rossi

Large Deviations for Clocks of Self-similar Processes

Abstract

The Lamperti correspondence gives a prominent role to two random time changes: the exponential functional of a Lévy process drifting to \(\infty\) and its inverse, the clock of the corresponding positive self-similar process. We describe here asymptotical properties of these clocks in large time, extending the results of Yor and Zani (Bernoulli 7, 351–362, 2001).

Nizar Demni, Alain Rouault, Marguerite Zani

Stochastic Bäcklund Transformations

Abstract

How does one introduce randomness into a classical dynamical system in order to produce something which is related to the ‘corresponding’ quantum system? We consider this question from a probabilistic point of view, in the context of some integrable Hamiltonian systems.

Neil O’Connell

The Kolmogorov Operator and Classical Mechanics

Abstract

The Kolmogorov operator is a quadratic differential operator which gives a typical example of a degenerate and hypoelliptic operator. The purpose of this note is to remark that the explicit expression for the transition probability density of the diffusion process generated by the Kolmogorov operator may be regarded as the Van Vleck formula. In fact, we show that it is given by the critical value of the action integral in some adequate path space.

Nobuyuki Ikeda, Hiroyuki Matsumoto

Explicit Formulae in Probability and in Statistical Physics

Abstract

We consider two aspects of Marc Yor’s work that have had an impact in statistical physics: firstly, his results on the windings of planar Brownian motion and their implications for the study of polymers; secondly, his theory of exponential functionals of Lévy processes and its connections with disordered systems. Particular emphasis is placed on techniques leading to explicit calculations.

Alain Comtet, Yves Tourigny

Matsumoto–Yor Process and Infinite Dimensional Hyperbolic Space

Abstract

The Matsumoto–Yor process is \(\int _{0}^{t}\exp (2B_{s} - B_{t})\,ds,t \geq 0\), where (B _t ) is a Brownian motion. It is shown that it is the limit of the radial part of the Brownian motion at the bottom of the spectrum on the hyperbolic space of dimension q, when q tends to infinity. Analogous processes on infinite series of non compact symmetric spaces and on regular trees are described.

Philippe Bougerol

Breadth First Search Coding of Multitype Forests with Application to Lamperti Representation

Abstract

We obtain a bijection between some set of multidimensional sequences and the set of d-type plane forests which is based on the breadth first search algorithm. This coding sequence is related to the sequence of population sizes indexed by the generations, through a Lamperti type transformation. The same transformation in then obtained in continuous time for multitype branching processes with discrete values. We show that any such process can be obtained from a d ²-dimensional compound Poisson process time changed by some integral functional. Our proof bears on the discretisation of branching forests with edge lengths.

Loïc Chaumont

Copulas with Prescribed Correlation Matrix

Abstract

Consider the convex set R _n of semi positive definite matrices of order n with diagonal \((1,\ldots,1).\) If μ is a distribution in \(\mathbb{R}^{n}\) with second moments, denote by R(μ) ∈ R _n its correlation matrix. Denote by C _n the set of distributions in [0, 1] ⁿ with all margins uniform on [0, 1] (called copulas). The paper proves that \(\mu \mapsto R(\mu )\) is a surjection from C _n on R _n if n ≤ 9. It also studies the Gaussian copulas μ such that R(μ) = R for a given R ∈ R _n . 

Luc Devroye, Gérard Letac

Remarks on the HRT Conjecture

Abstract

Motivated by a conjecture about time-frequency translations of functions, several properties of the Bargmann–Fock space \(\mathcal{H}\) and the Segal–Bargmann transform \(\mathcal{S}\) are investigated in this note. In particular, a characterization is given of those square integrable functions \(\varphi\) on \(\mathbb{R}\) such that \(z \in \mathbb{C}\longmapsto \mathcal{S}\varphi (z+\zeta ) \in \mathbb{C}\) is in \(\mathcal{H}\) for all \(\zeta \in \mathbb{C}\).

Daniel W. Stroock

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