At the intersection of probability theory (Markov processes, couplings) and analysis (PDE, functional inequalities), I study the long-time behavior of stochastic processes and their use in stochastic algorithms (such as MCMC or annealing), in particular in molecular dynamics. I am particularly interested in processes which are degenerated in some sense: non-reversible, hypoelliptic, hypocoercive, piecewise deterministic...
In the statistical physics framework, in order to conduct in silico experiments, a chemical system (a molecule, say) is considered as a random variable, distributed according to a Gibbs law associated to its energy. The quantities of interest, which are statistical means, are estimated through MCMC methods. There are several difficulties: high-dimensional and multiscales systems, metastability, complex geometry of the potential, etc., which ask for specific algorithms.
Sampling with memory
MCMC methods are based on the exploration of an unknown space by a Markov process, which by definition has no memory. Obviously, an amnesic explorer is not very efficient: it goes back many times to places it has already seen, and never learns anything. To improve that, without keeping in memory too much information, there are two solutions: either to keep a long-term memory of small dimension (ABF methods, metadynamics...), or keep a short (even instantaneous) memory, that is to say some inertia (kinetic processes).
In order, in particular, to check the convergence and compare the efficiency of algorithms, it is necessary to quantify the speed of convergence of stochastics dynamics toward their equilibrium (note that this question is much more general than just for algorithms). Yet, some classical tools (entropy methods, functional inequalities, etc.) which work well for, say, the overdamped Langevin process (a reversible elliptic diffusion), do not apply to kinetic ones (the kinetic Langevin diffusion, among others) or piecewise deterministc ones. For the latter, the convergence to equilibrium may be asymptotically exponential, but not with a constant rate: they are said hypocoercive.