Christian Léonard

Minimizers of energy functionals

We consider a general class of problems of minimization of convex integralfunctionals (maximization of entropy) subject to linear constraints. Undergeneral assumptions, the minimizing solutions are characterized. Our resultsimprove previous literature on the subject in the following directions: -anecessary and suficient condition for the shape of the minimizing densityis proved -without constraint qualification -under infinitely many linearconstraints subject to natural integrability conditions (no topological restrictions).As an illustration, we give the general shape of the minimizing density forthe marginal problem on a product space. Finally, a counterexample of I. Csiszáris clarified. Our proofs mainly rely on convex duality.

Large deviations and Nelson processes

We obtain a large deviation principle for the empirical process of a large dynamical stochastic System of non-interacting particles. A careful study of the rate function of this large deviation principle leads us to a construction of possibly degenerate difTusion processes with singular drifts. These processes play an important role in stochastic mechanics and are often called Nelson processes. 1991 Mathematics Subject Classification: 60F10, 81P20, 60G57, 60J60.

Minimization of entropy functionals

Entropy (i.e. convex integral) functionals and extensions of these functionals are minimized on convex sets. This paper is aimed at reducing as much as possible the assumptions on the constraint set. Primal attainment, dual equalities, dual attainment and characterizations of the minimizers are obtained with weak constraint qualifications. These results improve several aspects of the literature on the subject.

Convex conjugates of integral functionals

We consider a convex integral functional on a functional space V andcompute its greatest extension to the algebraic bidual space V**, among all convex functions which are lower semicontinuous with respect tothe *-weak topology o(V** ; V*).Such computations are usually performed to extend these functionals to sometopological closures. In the present paper, no a priori topological restrictionsare imposed on the extended domain. As a consequence, this extended functionalis a valuable first step for the computation of the exact shape of the minimizersof the conjugate convex integral functional subject to a convex constraint,in full generality: without constraint qualification. These convex integralfunctionals are sometimes called entropies, divergences or energies. Our proofsmainly rely on basic convex duality and duality in Orlicz spaces.

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

SummaryWe consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.

Reciprocal processes. A measure-theoretical point of view

This is a survey paper about reciprocal processes. The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal process. The structures of Markov and reciprocal processes are recalled with emphasis on their time-symmetries. A review of the main properties of the reciprocal processes is presented. Our measure-theoretical approach allows for a unified treatment of the diffusion and jump processes. Abstract results are illustrated by several examples and counter-examples.

Some epidemic systems are long range interacting particle systems

We present some recent results about dynamical interacting particle systems in the setting of epidemiology. The individuals are particles whose states (of health) depend on their relative positions. These individuals interact since they fall ill more often when their neighbours are infectious. We begin with a description of the interaction between two individuals at the microscopic level. Then we study the behaviour of the whole system at the macroscopic level, when the number of individuals tends to infinity.

Some Properties of Path Measures

We call any measure on a path space, a path measure. Some notions about path measures which appear naturally when solving the Schrodinger problem are presented and worked out in detail.

Large deviations for Poisson random measures and processes with independent increments

Large deviation principles are proved for rescaled Poisson random measures. As a consequence, Freidlin-Wentzell type large deviations results for processes with independent increments are obtained in situations where exponential moments are infinite.

Girsanov Theory Under a Finite Entropy Condition

This paper is about Girsanov’s theory. It (almost) doesn’t contain new results but it is based on a simplified new approach which takes advantage of the (weak) extra requirement that some relative entropy is finite. Under this assumption, we present and prove all the standard results pertaining to the absolute continuity of two continuous-time processes on \({\mathbb{R}}^{d}\) with or without jumps. We have tried to give as much as possible a self-contained presentation. The main advantage of the finite entropy strategy is that it allows us to replace martingale representation results by the simpler Riesz representations of the dual of a Hilbert space (in the continuous case) or of an Orlicz function space (in the jump case).