Dans cette thèse, on s'intéresse à l'étude d'existence de solutions pour les processus de rafle. Ce problème prend la forme d'une inclusion différentielle contrainte avec des cônes normaux qui apparaissent naturellement dans nombreuses applications telles que le mouvement de foule, l'élastoplasticité, les mécaniques, les circuits électroniques, etc. L'objective de ce travail est de rapprocher deux importantes classes d'inclusions différentielles. D'une part, nous établissons quelques résultats d'existence de tube-solutions pour des processus de rafle à des ensembles uniformément prox-réguliers. D'autre part, nous présentons des résultats d'existence de solutions monotone par rapport à un préordre pour un système mixte d'inclusions différentielles projetées. De plus, nous montrons l'existence d'un point-selle pour notre système et nous fournissons deux exemples d'applications. / In this thesis, we were interested in the study of the existence of solutions for sweeping processes. This problem takes the form of a constrained differential inclusion involving normal cones which appears naturally in many applications such as crowd motion, elastoplasticity, mechanics, electrical circuit, etc.The aim of this work is to bring together two classes of differential inclusions. On one hand, we establish some existence results of solutions-tube for sweeping processes with uniformly prox-regular sets. On the other hand, we present existence results of monotone solutions with respect to a preorder for a mixed system of projected differential inclusions. In addition, we show that our system has a saddle-point and we provide two examples of applications.

The goal of this thesis has been to throw light on a selection of open problems in chemical and biological physics using the general principles of statistical mechanics. These problems are all broadly concerned with the role of fluctuations in the dynamics of macromolecular systems. More specifically, they are concerned with identifying the microscopic roots of a number of interesting and unusual effects, including fractional viscoelasticity, anomalous chain cyclization dynamics in crowded environments, subdifffusion in hair bundles, symmetries in the work distributions of stretched polymers, heterogeneities in the geometries of reptation channels in polymer melts, and non-Gaussianity in the distributions of the end products of gene expression. I have shown here that all these effects are expressions of essentially the same underlying process of stochasticity, which can be described in terms of the dynamics of a point particle or a continuous curve that evolves in simple potentials under the action of white or colored Gaussian noise [8]. I have also shown that this minimal model of time-dependent behavior in condensed phases is amenable to analysis, often exactly, by path integral methods [13-15], which are naturally suited to the treatment of random processes in many-body physics. The results of such analyses are theoretical expressions for various experimentally measured quantities, comparisons with which form the basis for developing physical intuition about the phenomena under study. The general success of this approach to the study of stochasticity in biophysics and molecular biology holds out hopes of its application to other unsolved problems in these fields. These include electrical transport in DNA [143], quantum coherence in photosynthesis [144], power generation in molecular motors [145], cell signaling and chemotaxis [146], space dependent diffusion [147], and self-organization of active matter [148], to name a few. Most of these problems are characterized by non-linearities of one kind or another, so they add a new layer of complexity to the problems considered in this thesis. Although path integral and related field theoretic methods are equipped to handle such complexities, the attendant calculations are expected to be non-trivial, and the challenge to theory will be to devise effective approximation schemes for these methods, or to develop new and more sophisticated methods altogether.