# Stochastic processes, statistics and ergodic theory

There are three research groups working on stochastic processes, statistics and ergodic theory

## Stochastic processes

The members of the team are interested in studying random processes in various situations. Their work can be purely theoretic or motivated by the interaction between probability theory and other disciplines (analysis, geometry, statistics, physics, etc). Research themes cover a large spectrum from process analysis to interactions with analysis, geometry and statistical applications: stochastic partial derivatives equations (SPDE), backward stochastic differential equations (BSDE), non-Markovian processes, piecewise deterministic Markovian processes (PDMP), Lévy processes, stochastic differential geometry, limit theorems,... The team organizes a weekly seminar and a workshop every Monday.

## Statistics

Statistics of complex data and big data

The current evolution of statistics is strongly influenced by the increasing diversity of data formats and the inflation of their volume. It focuses analytical approaches on systemic approaches, based on the idea that data at different scales and natures provide complementary points of view on the system studied. Here are some of the challenges:

• Rethink the models and the notion of inference to understand the dependencies between variables and phenomena studied for high-dimensional data.
• Taking into account the variety of the data, linked to the nature of the data as well as to their selection mechanism (sampling design, non-response) or to their resolution, often assimilated to the signal-to-noise ratio.
• Adaptation of data visualization techniques to the data volume.
• Adaptation of algorithms to high-dimensional data.

Statistical process

• Modeling and statistical inference for inhomogeneous temporal data when the parameters of the stochastic process evolve over time.
• Study of the behavior of extreme values ​​for locally stationary processes and censored lifetime processes.
• Estimation of jump rates to different classes of deterministic Markov processes.
• Study of determining processes (test of the determinant nature of a process, estimation, etc.).

## Ergodic theory

The research area of the ergodic theory team is in the interface of dynamics, probability and geometry. It also has close connections with that of other teams in the IRMAR such as stochastic processes, probability, analytic geometry and arithmetic geometry. To mention a few keywords, let us highlight the work of Bailleul on rough paths, that of Cantat and Xie on birational dynamics, that of Maucourant, Shapira and Bekka on dynamics on homogenous spaces and finally that of Guirardel and Coulon on geometric group theory. It goes without saying that this list is not exhaustive. For example one should also definitively mention the work of Guivarch and Conze on the asymptotic behaviour of random walks.

The variety of topics of covered by the members of the ergodic theory team is reflected in the journals in which their work is published. Besides general mathematics journals such as for example Acta. Math, Duke, J. AMS, Crelle, or Ann. Sci. ENS, there have been numerous publications in Ergodic Theory and Dynam. Systems, Geometry and Topology, the Journal of Topology, GAFA, J. Functional Anal, Annals of Probability, Nonlinearity, SIAM J. Optimization and Control, SIAM J. on Numerical Analysis, or SIAM J. on Scientific Computing.

In some sense, the broadness of the area of research covered by the ergodic theory team is one of its strengths because it leads to collaborations between researchers with different
sensibilities. The team runs a seminar as well as a reading group. The later has covered topics such as random walks on groups, buildings, or aspects of geometry and dynamics on hyperbolic manifolds. The reading seminar serves not only for the members of the team acquire some specialised knowledge, but also strengthens the cohesion of the team, and plays an important role in the training of graduate students.

The training of graduate students plays an important role in the life of
the team: currently 13 permanent members has passed their HDR (or equivalent), there are about as many graduate students, and every year several of the members of the team give courses in either the graduate student of the master level.