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Statistical Laboratory - DPMMS, Tel: (+44)01223 766925 |
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Statistical Laboratory - DPMMS, Tel: (+44)01223 766925 |
Welcome to my webpage! I'm currently a research associate in the Statistical Laboratory, in Cambridge. I'm working here on coagulation problems and relativistic diffusions. (Click here for a brief CV .)
I am teaching this year (Michaelmas 2010) the Part III course Advanced Probability. You can find the lecture notes for the course below; it does not replace the course in any case and is to be use with care, as typos or mistakes may have escaped my careful reading.
My main research area is at the intersection of probability, geometry and mathematical physics, and is concerned with the study of random processes intrinsically associated with any relativistic spacetime model. I have also studied (the homogeneous and non-honogeneous version of) Smoluchowski equation giving the macroscopic evolution of some interacting particle systems. Spacetime random dynamics and particle systems are the main two ingredients of the study of some nonlinear processes on Lorentzian manifolds, in relation with the relativistic Boltzmann equation.
Smoluchowski equation is a model of dynamics where clusters of different species coagulate at a rate depending on their characteristics. This model encompasses a large number of practical situations, from chemical reactions to powder storage... It is an important matter to understand how solutions of this equation depend on the parameters of the dynamics. The derivative of the solution with respect to the parameter is called the sensitivity. As a first step towards a detailled study of this dependence, I prove in this article that the measure-valued function representing the state of the system is a C1 function of these parameters, in a good space of measures.
The second article (written M. Kraft and P. Man) is the numerical counterpart of the above one. The theoretical result obtained there suggests a Marcus-Lushnikov like particle system to approximate the sensitivity. Convergence of the particle system is proved and numerical experiments are performed, showing the great accuracy and low variance of the estimator.
The third article (written with M. Kraft, P. Man and J. Norris) presents an alternative method to simulate the sensitivity in which a coupled pair of Marcus-Lushnikov process is used to simulate two solutions to Smoluchowski equation with close parameters.
I prove that the spatial coagulation equation with bounded coagulation rate is well-posed for all times in a given class of kernels if the convection term of the underlying particle dynamics has divergence bounded above by a negative constant. Multiple coagulations, fragmentation and scattering are also considered.
In the same way as Brownian motion is the only continuous strong Markov process in the Euclidean space Rn whose law is invariant by the action of the isometries, there is essentially a unique way to define a C1 random path in the space of special relativity, representing the motion of an object having a speed less than the speed of light, and whose law is invariant by the action of the isometries of this space. The study of the asymptotic behaviour of this unique object, called "relativistic diffusion", is lead in these two articles using two different approaches. Whereas the first uses SDEs, stochastic calculus and couplings, the second (written with Albert Raugi) restates the problem in terms of the asymptotic behaviour of a random walk on a non semi-simple Lie group and highlights the deep geometric nature of the result.
Following the idea of C. Chevalier and F. Debbasch presented in their article 'Relativistic diffusions: a unifying approach', J. Math. Phys. 49 (4), 2008, we consider a class of relativistic diffusions, roughly characterized by the fact that there exists at each (proper) time (of the moving particle) a (local) rest frame where the acceleration of the particle is Brownian in any spacelike direction of the frame, when computed using the time of the rest frame. These diffusions are called by the authors relativistic diffusions. A pathwise approach of these processes is proposed here, in the general framework of Lorentzian geometry. The results proved not only provide a dynamical justification of the analytical approach developped up to now, and a new general H-theorem, it also sheds some light on the importance of the large scale structure of the manifold in the asymptotic behaviour of the Franchi-Le Jan process, studied in the above articles in Minkowski spacetime.
In so far as relativistic diffusions are defined in purely geometric terms, it is very likely that part (or all?) of the geometry of the ambient spacetime may be recovered from the probablistic behaviour of these processes. In a Riemannian setting, this probabilistic view on geometry is well-illustrated by Weyl and Pleyel formulas for the heat kernel of Brownian motion where local and global informations about the geometry appear. We investigate in this work one aspect of this geometry/probability correspondence. Dating back to Penrose and Hawking's results, it is now well-established that the appearance of singularities in Einstein's theory of gravitation is unavoidable under quite natural assumptions. Although there is no definitive agreement on what should be called a singularity of spacetime, a largely used notion of singularity is the existence in spacetime of incomplete geodesics. Is there a link between geodesic and probabilistic incompleteness? This work concentrates on one aspect of this question and provides different criteria under which the diffusion does not explode. Joint work with J. Franchi.
The aim of this work is to promote the use of probabilistic methods in the study of problems in mathematical general relativity. Two new singularity theorems, whose features are different from the classical singularity theorems, are proved using probabilistic methods. Under some energy conditions, and without any causal or initial/boundary assumption, simple conditions on the energy flow imply probabilistic incompleteness. Also we introduce a probabilistic notion of spacetime boundary which has none of the pathological defects that the classical boundaries may have.
Following classical works initiated by Tanaka in the seventies in the framework of the space homogeneous Boltzmann equation, we show how one can associated intrinsically to the general relativistic version of Boltzmann equation a Markov process. It is used to prove the causal character of the general relativistic Boltzmann equation on strongly causal spacetimes.
In this note, I give a short proof of a result of Dufresne on some exponential functional of a geometric Brownian motion.
This note shows how to extend Benjamini and Schramm's one dimensional KPZ identity in a mutlidimensional random geometry of multiplicative cascades.
A link to my PhD thesis on the Poisson boundary of the relativistic diffusion in Minkowski spacetime (December 2006).
You can find below three lecture notes that I have written about different aspects of potential theory, at a basic level; they are written in french. The first paper is about the probabilistic treatment of Dirichlet problem, the second is about electrostatics and equilibrium measures (without probability), and the third one is about convexity, from the basics to Krein-Milman theorem and Choquet theory on integral representations in convex sets.