Stochastic Processes, Ergodic Theory and Stochastic Modeling


Researchers: Joaquín FontbonaRaúl GouetAlejandro MaassServet Martínez, Daniel RemenikJaime San MartínMichael Schraudner
Postdocs: Claire Delplancke, Fabio Lopes, Nga Nguyen

About the research group

Our group works on a variety of fundamental topics in probability theory, stochastic processes, statistical physics, and ergodic theory. We also develop applications to a diverse range of subjects where randomness plays a key role, including systems biology and bioinformatics, astroinformatics, mining, renewable and non-renewable resources, and simulation and statistical data analysis for engineering and finance.

Quasi-stationary distributions and M-matrices are two areas of long-standing interest for the group, which have led to two recent Springer monographs describing the state-of-the-art of these theories. The group is currently focused on specific problems stemming from these subjects, including applications in ecology.

Another focus of work is stochastic models in mathematical physics. One area of interest is the KPZ universality class. A highlight of our research in this subject has been the recent construction of the KPZ fixed point, the universal Markov process describing the asymptotic spatial fluctuations of all models in the class, which opens several lines of future research. Another focus of interest for the group is the long time behavior of interacting particle systems. We have made important contributions to the development of probabilistic techniques for establishing propagation of chaos for mean field models in kinetic theory, such Kac particle systems associated with the Boltzmann equation.

In subjects closer to mathematical statistics, we have contributed to the study of records, including theoretical results for i.i.d. random variables and the development of new inferential procedures for the geometric distribution based on δ-records, and we have introduced the method of Geodesic Principal Component Analysis (GPCA), which presents several advantages over the functional PCA approach. Other work directly inspired by applied modeling includes the study of some aspects of the probabilistic structure of bacterial DNA related to Chargaff’s second parity rule and the use of entropy minimization techniques in the setting of optimization in financial markets under model uncertainty. We have also developed work on Ray-Knight theorems for Lévy trees and the associated continuum genealogical description for flows of branching processes with competition.

In ergodic theory an important focus has been the study of recurrence and local complexity, in particular in the setting of dynamics of nil systems, which play an important role in the study of various problems in number theory and combinatorics. A more recent interest of our group is in symbolic dynamics related to the automorphism groups of minimal Cantor systems. We have also contributed to the spectral theory of low complexity symbolic systems and models of quasicrystals in physics; we have provided explicit necessary and sufficient conditions for a Toeplitz system of finite topological rank to have measure theoretical but not continuous eigenvalues. In multidimensional symbolic dynamics we have introduced the novel notion of projective subdynamics of Zd subshifts and obtained several related results; we have also worked on the realizability of topological entropies for block gluing Zd shifts of finite type and on a combinatorial characterization of the dynamical notion of entropy minimality for Zd subshifts of finite type.