Probabilités en physique et biologie numérique: quelques contributions mathématiques.

The scientific activity presented within this thesis is mainly dedicated to the mathematical study of models coming from computational physics, and to a less extent, biology. These models are the following:(i) Stochastically perturbed (thermostatted) Hamiltonian systems. Such systems are widely used in classical molecular simulation.(ii) Fermionic Schr\"odinger operators, which describes non-relativistic systems of Fermionic (exchangeable) particles, and are central to computational chemistry (the particles are the electrons of molecules).(iii) Individual-based models of bacterial chemotaxis. Such models describe the random motion of each bacterium depending on the chemical environment.(iii) Boltzmann's kinetic theory of rarefied gases; with focus on the space homogeneous simplification, as well as on the associated conservative particle systems.Although the material is mainly written in mathematical style, the physics of the considered systems is a source of motivation and intuition. Some contributions are merely theoretical, with mathematical theorems analyzing some physically relevant features of the models. Other contributions are more applied, with suggestions of numerical methods and realistic numerical tests.Different standard mathematical tools are required in the basic analysis of the considered problems. For instance, some concepts of differential geometry for Hamiltonian systems with constraints; the spectral theorem for Schrödinger operators; and usual stochastic calculus associated with Markov processes.In the same way, the classical spectrum of probabilistic tools are used in the core of the presented contributions. For instance, the reader which is not an expert in probability theory shall not be completely unfamiliar with the following concepts: (i) changes of probability measures for stochastic processes, (ii) tightness and convergence in (probability distribution) law for processes; (iii) basic Feynman-Kac representation of parabolic partial differential equations, probabilistic interpretation of boundary conditions; (iv) Coupling methods, which may be summarized for the unfamiliar reader with the aphorism: ``the same random numbers are used to construct (or simulate) and compare two similar systems, or models''