La recherche

In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltoni-ans by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity: a property is generic if, given a Hamiltonian H, the set of potentials u such that H + u satisfies the property is generic. This notion is mostly used in the context of Hamiltonians which are convex in p, in the sense that ∂ 2 pp H is positive definite at each points. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians H + u and H/(1 − u) have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when H is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be perturbed by adding a small potential. This kind of question depend strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, Mañé perturbations of convex Hamiltonians. It is for example well-known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka-Smale theorem, see [5, 13]. There is no such result in the context of Hamiltonian vector fields, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [4, 2, 1]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [9, 10]. It is proved in [10] that the same result holds : If H is a convex Hamiltonian and a is a regular value of H, then there exist arbitrarily small potentials u such that all periodic orbits (including iterated ones) of H + u at energy a are non-degenerate. The proof given in [10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work

We study a hard sphere gas at equilibrium, and prove that in the low density limit, the fluctuations converge to a Gaussian process governed by the fluctuating Boltzmann equation. This result holds for arbitrarily long times. The method of proof builds upon the weak convergence method introduced in the companion paper [8] which is improved by considering clusters of pseudo-trajectories as in [7].

Throughout this PhD thesis we will study two probabilistic objects, Gaussian multiplicative chaos (GMC) measures and Liouville conformal field theory (LCFT). The GMC measures were first introduced by Kahane in 1985 and have grown into an extremely important field of probability theory and mathematical physics. Very recently GMC has been used to give a probabilistic definition of the correlation functions of LCFT, a theory that first appeared in Polyakov's 1981 seminal work, "Quantum geometry of bosonic strings". Once the connection between GMC and LCFT is established, one can hope to translate the techniques of conformal field theory in a probabilistic framework to perform exact computations on the GMC measures. We start from the BPZ equations for LCFT, introduced by Belavin, Polyakov and Zamolodchikov in 1983. The mechanism of these equations is studied in the last part of this thesis and we prove the higher order BPZ equations with a general formalism. Following the probabilistic methods established by Kupiainen-Rhodes-Vargas for the resolution of the BPZ equations and after overcoming several major difficulties, we obtain non trivial relations for some fundamental objects of LCFT. More precisely, we prove the exact formulas for all the four structure constants of LCFT on the disk with null cosmological constant in the bulk, one of which was solved by Remy in 2017. As a special case, we find the distribution of the total mass of GMC on the interval with log-singularities put on both ends, a conjecture that has been independently predicted by Ostrovsky and by Fyodorov, Le Doussal, and Rosso in 2009. Another direct consequence is the law of the total mass of GMC on the unit circle with a log-singularity, conjectured by Ostrovsky in 2016.