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Gabriel Faraud
I am currently a research fellow at
the WIAS Berlin
(Research Group "Interacting Random systems"), I am also a member of
the ANR project "MEMEMO"
Contact
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Gabriel Faraud
I am currently a research fellow at
the WIAS Berlin
(Research Group "Interacting Random systems"), I am also a member of
the ANR project "MEMEMO"
Contact
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Gabriel Faraud
Studies
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Research Interest
I defended on September, the 3rd 2010 my Phd thesis entitled On some random processes in a random environment. This thesis has been prepared at the Laboratoire d'Analyse, Géométrie et Applications de l'Université Paris 13 under the direction of Yueyun Hu. My work during this preparation has lead me to take interest into various problems in the domain of Random Processes in a random environment, in particular the case of Random Walk in a Random Environment (RWRE) on trees, and the continuous-time analogue of RWRE known as Brox's Diffusion. Apart from the classical techniques used in the RWRE's litterature, this has been for me the occasion to discover many tools in probability theory, such as the relationship between electric networks and random walks on graphs introduced by P.G. Doyle and J.L. Snell, moderate and large deviations methods, martingale and moments inequalities and several techniques used to prove central limits theorems for Markov chains, as well as continuous time techniques such as stochastic Calculus, Stochastic Differential Equation, Ray-Knight and Girsanov's Theorems, Bessel Process and Bessel Bridges studies, spectral Analysis... More recently, while still working on the domain of random processes in random environment, I extended my interest to models of interacting particle systems, in particular in some applications to communication networks, in collaboration with Hanna Döring and Wolfgang König. I have also much "non-professional" interest in several research areas less connected to my own research such as Large Random Matrices, Potential Theory and in general any area of mathematics related to Physics. Published articles
Estimates
on the Speedup and Slowdown for a Diffusion in a Drifted Brownian
Potential We study a model of diffusion in a Brownian potential. This model was first introduced by T. Brox (Ann. Probab. 14:1206–1218, 1986) as a continuous time analogue of random walk in random environment. We estimate the deviations of this process above or under its typical behavior. Our results rely on different tools such as a representation introduced by Y. Hu, Z. Shi and M. Yor, Kotani’s lemma, introduced at first by K. Kawazu and H. Tanaka (J. Math. Soc. Jpn. 49:189–211, 1997), and a decomposition of hitting times developed in a recent article by A. Fribergh, N. Gantert and S. Popov (Preprint, 2008). Our results are in agreement with their results in the discrete case. A
central limit Theorem for random walk in a random environment on a
marked Galton-Watson tree Models of random walks in a random environment were introduced at first by Chernoff in 1967 in order to study biological mechanisms. The original model has been intensively studied since then and is now well understood. In parallel, similar models of random processes in a random environment have been studied. In this article we focus on a model of ran- dom walk on random marked trees, following a model introduced by R. Lyons and R. Pemantle (1992). Our point of view is a bit different yet, as we consider a very general way of constructing random trees with random transition probabilities on them. We prove an analogue of R. Lyons and R. Pemantle’s recurrence criterion in this setting, and we study precisely the asymptotic behavior, under restrictive assumptions. Our last result is a generalization of a result of Y. Peres and O. Zeitouni (2006) concerning biased random walks on Galton-Watson trees. Almost
sure convergence for stochastically biased random walks on trees We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)^3, converges almost surely to a known positive constant. Bessel bridges
decomposition with varying dimension. Applications to finance We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at $t$ of the process and its integral. We finally give some financial applications of our results. Connection times in large ad-hoc mobile networks
We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances lesser than 2R, with 2R the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model. We consider the connection time of two sample participants, the amount of time over which these two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random, and a global, deterministic mechanism, and we give a formula for the limiting behaviour. A prime example of the movement schemes that we consider is the well-known random waypoint model (RWP). Here we describe the decay rate, in the limit of large time horizons, of the probability that the portion of the connection time is less than the expectation. |
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Teaching
At the université Paris 13
Outside the UniversityI participated between 2007 and 2010 to the Tutorat , organized by Farouk Bouccekine in collaboration with Animath, Science Ouverte, the APMEP and the ENS. |