Topics in stochastic analysis

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Probability theory is the mathematical theory concerned with the analysis of random phenomena. Many of such phenomena may be modeled by continuous time stochastic processes. The first stochastic process that has been extensively studied is the celebrated Brownian motion, named in honor of the botanist Robert Brown, who observed and described in 1828 the random movement of particles suspended in a liquid or gas. The same process was later used in 1900 by the mathematician Louis Bachelier to model stock prices on financial markets. Finally, in 1905, Albert Einstein brought this process to the attention of physicists by presenting it as a way to indirectly confirm the existence of atoms and molecules. The Brownian motion is an example of diffusion process. Like their ancestor the Brownian motion, diffusion processes appear in many different areas of sciences and economy and their theoretical mathematical study has far reaching consequences in understanding and making predictions about the phenomena they model. In this project, the PI will study several problems in the theory of diffusion processes. In particular, questions about the deep interaction between the diffusion process and the geometry of the ambient space will be addressed and rates of convergence to equilibrium will be studied.Mathematically speaking, the present project focuses on different aspects of the theory of diffusion processes and diffusion semigroups. The PI will investigate applications to sub-Riemannian geometry where diffusion methods turn out to be very fruitful to study generalized Ricci curvature lower bounds. The PI will also address several questions about hypocoercive diffusions. Hypocoercivity is a concept recently introduced by Cedric Villani to obtain quantitative estimates for the convergence to equilibrium of some highly degenerate hypoelliptic semigroup. Hypocoercive estimates are in general very difficult to prove and the PI will systematically study new methods which parallel the Bakry-Emery approach to hypercontractivity. Some problems in the rough paths theory of Terry Lyons will also be studied, in particular related to the properties of stochastic differential equations driven by Gaussian processes. These projects will involve the training of graduate students and junior mathematicians. The results will be disseminated through publications in professional journals, lectures and on the blog of the PI.