Contributions to the study of biological models, functional inequalities, and random matrices.

Summary The work presented concerns three autonomous themes: (1) Biological and statistical models: compartmental models, population pharmacokinetics and pharmacodynamics, estimators for stochastic inverse problems, nonlinear mixed-effects models, mixture models, EM and ICF algorithms, graphical covariance models, cancer modeling, point processes, particles, queues, renormalization of inhomogeneous Markov processes and Feynman-Kac formulas (2) Functional inequalities: Sobolev-type inequalities, measure concentration, isoperimetry role of convexity in entropic inequalities, tensorization, heat kernel, Heisenberg group and hypoelliptic dynamics, queues, mixtures of laws (3) Random matrices: spectrum of random Markovian matrices, random weight graphs, Wigner, Marchenko-Pastur, and Girko-Bai type theorems, convergence of extremal eigenvalues, rank-one deformations. The most recurrent concept here is that of Markovian dynamics. In the first part, compartmental models of pharmacology are related to such dynamics. The second part deals with functional inequalities associated with the speed and geometry of Markov dynamics. Finally, the third part deals with random Markov dynamics. These three parts are not reduced to the study of facets of Markovian problems. Their content covers a spectrum that is both theoretical and applied, and uses various techniques and concepts from analysis, probability, and statistics.