Papers by Olivier Raimond
The Annals of Probability
We are interested in stationary "fluid" random evolutions with independent increments. Under some... more We are interested in stationary "fluid" random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
Nous présentons une partie des résultats de Le Jan et Raimond (math.PR/0203221). Nous montrons co... more Nous présentons une partie des résultats de Le Jan et Raimond (math.PR/0203221). Nous montrons comment à partir d'une famille compatible de semigroupes felleriens, on peut construire un flot stochastique de noyaux. Sous une hypothèse supplémentaire (sur le mouvement de deux points), nous montrons qu'à un flot de noyaux, il est possible d'associer un flot coalescent tel que le flot de noyaux puisse être construit en filtrant ce flot coalescent par un sous-bruit d'une extension du bruit engendré par le flot coalescent. Pour citer cet article : Y. Le Jan, O. Raimond, C. R. Acad. Sci. Paris, Ser. I 336 (2003). 2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.
Comptes Rendus Mathematique
Comptes Rendus Mathematique
We show that the only flow solving the stochastic differential equation (SDE) on R
Illinois journal of mathematics
Latin American journal of probability and mathematical statistics
We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph wi... more We define a Tanaka's equation on an oriented graph with two edges and two vertices. This graph will be embedded in the unit circle. Extending this equation to flows of kernels, we show that the laws of the flows of kernels K solution of Tanaka's equation can be classified by pairs of probability measures (m + , m − ) on [0, 1], with mean 1/2. What happens at the first vertex is governed by m + , and at the second by m − . For each vertex P , we construct a sequence of stopping times along which the image of the whole circle by K is reduced to P . We also prove that the supports of these flows contains a finite number of points, and that except for some particular cases this number of points can be arbitrarily large.
Latin American journal of probability and mathematical statistics
ABSTRACT
Latin American journal of probability and mathematical statistics
We study a natural continuous time version of excited random walks, introduced by Norris, Rogers ... more We study a natural continuous time version of excited random walks, introduced by Norris, Rogers and Williams about twenty years ago. We obtain a necessary and sufficient condition for recurrence and for positive speed. This is analogous to results for excited (or cookie) random walks.
Latin American journal of probability and mathematical statistics
We study Vertex-Reinforced-Random-Walk on the complete graph with weights of the form $w(n)=n^\al... more We study Vertex-Reinforced-Random-Walk on the complete graph with weights of the form $w(n)=n^\alpha$, with $\alpha>1$. Unlike for the Edge-Reinforced-Random-Walk, which in this case localizes a.s. on 2 sites, here we observe various phase transitions, and in particular localization on arbitrary large sets is possible, provided $\alpha$ is close enough to 1. Our proof relies on stochastic approximation techniques. At the end of the paper, we also prove a general result ensuring that any strongly reinforced VRRW on any bounded degree graph localizes a.s. on a finite subgraph.
Stochastic Processes and their Applications, 2015
We study a stochastic differential equation (SDE) driven by a finite family of independent white ... more We study a stochastic differential equation (SDE) driven by a finite family of independent white noises on a star graph, each of these white noises driving the SDE on a ray of the graph. This equation extends the perturbed Tanaka's equation recently studied by Prokaj [16] and Le Jan-Raimond [11] among others. We prove that there exists a (unique in law) coalescing stochastic flow of mappings solution of this equation. Our proofs involve the study of a Brownian motion in the two dimensional quadrant obliquely reflected at the boundary, with time dependent angle of reflection. Filtering this coalescing flow with respect to the family of white noises yields a Wiener stochastic flow of kernels also solution of this SDE. This Wiener solution is also unique. Moreover, if N denotes the number of rays constituting the star graph, the Wiener solution and the coalescing solution coincide if and only if N = 2. When N ≥ 3, the problem of classifying all solutions is left open. Finally, we define an extension of this equation on more general metric graphs to which we apply some of our previous results . As a consequence, we deduce the existence and uniqueness in law of a flow of mappings and a Wiener flow solutions of this SDE.
The IMA Volumes in Mathematics and its Applications, 2005
ABSTRACT
The Annals of Probability, 2002
Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.'s ... more Using the Wiener chaos decomposition, we show that strong solutions of non Lipschitzian S.D.E.'s are given by random Markovian kernels. The example of Sobolev flows is studied in some detail, exhibiting interesting phase transitions.
SIAM Journal on Control and Optimization, 2010
This paper studies a class of non-Markovian and nonhomogeneous stochastic processes on a finite s... more This paper studies a class of non-Markovian and nonhomogeneous stochastic processes on a finite state space. Relying on a recent paper by Benaïm, Hofbauer, and Sorin [SIAM J. Control Optim., 44 (2005), pp. 328-348] it is shown that, under certain assumptions, the asymptotic behavior of occupation measures can be described in terms of a certain set-valued deterministic dynamical system. This provides a unified approach to simulated annealing type processes and permits the study of new models of vertex reinforced random walks and new models of learning in games such as Markovian fictitious play.
Probability Theory and Related Fields, 2012
We obtain the convergence in law of a sequence of excited (also called cookies) random walks towa... more We obtain the convergence in law of a sequence of excited (also called cookies) random walks toward an excited Brownian motion. This last process is a continuous semi-martingale whose drift is a function, say ϕ, of its local time. It was introduced by Norris, Rogers and Williams as a simplified version of Brownian polymers, and then recently further studied by the authors. To get our results we need to renormalize together the sequence of cookies, the time and the space in a convenient way. The proof follows a general approach already taken by Tóth and his coauthors in multiple occasions, which goes through Ray-Knight type results. Namely we first prove, when ϕ is bounded and Lipschitz, that the convergence holds at the level of the local time processes. This is done via a careful study of the transition kernel of an auxiliary Markov chain which describes the local time at a given level. Then we prove a tightness result and deduce the convergence at the level of the full processes.
Probability Theory and Related Fields, 2004
This paper gives a construction of sticky flows on the circle. Sticky flows give examples of stoc... more This paper gives a construction of sticky flows on the circle. Sticky flows give examples of stochastic flows of kernels that interpolates between Arratia's coalescing flow and the deterministic diffusion flow. They are associated with systems of sticky independent Brownian particles on the circle, for some fixed parameter of stickyness.
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Papers by Olivier Raimond