Markov processes and variational problems

In this thesis, we study the statistical properties of non-linear transforms of Markov processes.These transforms are defined via variational formulas, and arise in various fields such as statistics, mathematical finance, convex analysis, statistical mechanics and hydrodynamic turbulence. In particular, we will focus on two sets of problems. The first problem that is addressed in Chapter 2, concerns the study of the Lipschitz minorant of the sample paths of a L´evy process. The study of this minorant was initiated by Abramson and Evans, but here we shed a light on its excursion structure away from its contact set. When the L´evy process is a Brownian motion with drift, an explicit path decomposition of these excursions is given, together with the decomposition of the semimartingales in the progressive enlargement of the canonical filtration by the first positive point in the contact set. In the second set of problems, we will consider physical solutions (also called entropy solutions) to scalar conservation laws (or equivalently Hamilton-Jacobi equations) with random initial data. In Chapter 3, we investigate the distribution of the solutions at later times in the one-dimensional case and when the initial data is a Brownian white noise for any general convex Hamiltonian. This settles a conjecture of Menon and Srinivasan and extends Groeneboom’s result for the Burgers case. In Chapter 4, we consider the higher dimensional case and focus in particular on the planar case. We construct a family of random convex piecewise linear functions which are the dimension 2 analogue of the anti-derivative of pure-jump Markov processes. At the heart of this construction is a novel class of kinetic equations. The invariance of this class of processes under the flow of Hamilton-Jacobi equations is also discussed