Stochastic Processes, Ergodic Theory and Stochastic Modeling

Researchers

CMM: Joaquín FontbonaRaúl GouetAlejandro MaassServet MartínezJaime San MartínMichael Schraudner

U. Concepción: Carlos Mora

UAI: Javiera Barrera 

UTFSM: Ronny Vallejos 

UNAB: José Aliste, Daniel Coronel, Pierre Paul Romagnoli

About the research group

Randomness plays a key role in statistical mechanics, mathematical finance, computer science and bioinformatics, all of which are areas of our interest. We work on a variety of fundamental topics like killed processes, local entropy, thermodynamic formalism, spectral theory of dynamical systems, random fragmentation, and simulation of interacting systems. Applications range from systems biology to mining and astronomy.
The main theoretical research topics of our group are stochastic processes, ergodic theory, particle systems, mathematical physics and probability in discrete structures. The applications and our research in applied modeling are mainly concentrated on bioinformatics applications in renewable and non-renewable resources, simulation and statistical data analysis for engineering and astro-informatics; three of our main researchers are the directors of the CMM laboratories devoted to each of these three subjects (respectively A. Maass, J. Fontbona and Ja. San Martín) and a fourth one directs a project related to marketing mix optimization (D. Remenik). These laboratories attract bright young scientists, maintain contracts with the mining, aquiculture, agronomical and other industries, and have interactions with many other institutions such as CTIO, ALMA, ESO and numerous universities in USA, Europe and Japan.

Recurrence and local complexity in ergodic theory.

In earlier years we have contributed to the understanding of the structure of nilfactors of a topological dynamical system, and in this period we have continued to study different aspects of the dynamics of nil systems . This class of systems has shown to play an important role in the study of different problems in number theory and combinatorics. We have obtained the explicit polynomial complexity for nilsystems and used this result to understand some classical ergodic theorems in systems where nil factors of different degrees coincide. We have also proved a partial result related to the Katznelson conjecture, showing that any recurrence set for rotations is a recurrence set for any nil system and its proximal extensions; we also discovered a new phenomenon in recurrence proving that a set of s-multiple recurrence for an s-step nil system is a set of s-recurrence for any t-step nil system for t≥s. We have also started work in the new and hot topic in symbolic dynamics aiming to describe the automorphism groups of minimal Cantor systems. We have proved that the automorphism group of a minimal system of linear complexity is virtually the integers and provide many interesting examples on the same topic for arbitrary polynomial complexity. Finally, we have proved that order 2 nilsystems are characterized by the order 2 nilpotence of the associated enveloping group and that the enveloping system of an order d nilsystem is nilpotent with order d. This result follows from a fine application of the cube structures developed by Host-Kra-Maass.

Low complexity systems and minimal subshifts.

We have contributed to the spectral theory of low complexity symbolic systems and models of quasicrystals in physics in recent years. We have provided explicit necessary and sufficient conditions for a Toeplitz system of finite topological rank to have measure theoretical but not continuous eigenvalues.

Multidimensional symbolic dynamics.

We introduced the novel notion of projective subdynamics of Zd subshifts for which we obtained various results. Recently, we obtained a complete classification in the case where the Zd shift of finite type satisfies some kind of uniform mixing condition, and they obtained a complete classification in the case where the projective subdynamics is a one- dimensional sofic system. Another work studied the realizability of topological entropies for block gluing Zd shifts of finite type. Finally, we gave a combinatorial characterization of the dynamical notion of entropy minimality for Zd subshifts of finite type, which allowed them to determine the support of measures of maximal entropy in certain families of Zd SFTs.

Probabilistic structure in bacterial DNA.

We have studied the probabilistic structure of sequences of START and STOP codons in bacterial DNA, which represent the boundaries between coding and non-coding regions. We showed that a form of Markovianness, as well as a conditional independence property, is present in this sequence, and that it furthermore satisfies Chargaff’s second parity rule.

Quasi-stationary distributions.

Continuing our work in the area over the last several years, which led to the publication of a monograph on the subject in 2013, we studied a Markov process evolving in the natural numbers with fast return from infinity, which are conditioned not to hit 0. We have proved that the process converges exponentially fast to its quasi-stationary distribution. The result has applications to birth and death processes and a class of possibly non-irreducible processes.

STIT tessellations.

We studied homogeneous STIT tessellations, which correspond to certain Markov processes in the space of tessellations of the Euclidean space. We have proved that the tail σ-algebra associated to the process is trivial (Adv. Appl. Probab. 2014), which constitutes a sharpening of earlier mixing results by other authors. Later on, we showed that if (Yt) is a STIT tessellation process and a>0 then the process (α(t)Yα(t)), with α(t)=at, is a Bernouilli flow, and that the filtration associated to the discrete time version of the last process is standard.

Inverse M-matrices and ultrametric matrices.

During 2014 we published the book Inverse M-Matrices and Ultrametric Matrices in the Springer collection Lecture Notes in Mathematics.
This work is related to our earlier research in ultra metric potential. The main focus of the book is the so-called inverse M-matrix problem, which asks for a characterization of nonnegative matrices whose inverses are M-matrices. An answer is provided in terms of discrete potential theory based on the Choquet-Deny Theorem. A distinguished subclass of inverse M-matrices is that of ultrametric matrices, which are important in applications to fields such as taxonomy. Ultrametricity is revealed to be a relevant concept in linear algebra and discrete potential theory because of its relation with trees in graph theory and mean expected value matrices in probability theory.

Records.

We worked on several problems related to the study of records and near-records. We developed new inferential procedures for the geometric distribution, based on δ-records, including maximum likelihood and Bayesian approaches for parameter estimation and prediction of future records. The performance of the estimators is compared with those based solely on record-breaking data by means of Monte Carlo simulations, concluding that the use of δ-records is clearly advantageous. In separate work, we considered the point process formed by the near-record values of a sequence of i.i.d. random variables, showing that it is given by a Poisson cluster process. We also provided characterizations as well as a study of the convergence of this process.

Geodesic PCA.

We introduced the method of Geodesic Principal Component Analysis (GPCA) on the space of probability measures on the line with finite second moment, endowed with the Wasserstein metric. The advantages of this approach over a standard functional PCA and many properties and particular examples were discussed.

Long time behavior of interacting particle systems and SDEs.

We studied several problems related to the convergence and long time behavior of interacting particle systems and related processes. We established exponential convergence rates to equilibrium for mean field stochastic vortex systems that tend to the two-dimensional Navier-Stokes equation as the number of particles gets large. In related work, we established sharp propagation of chaos estimates for binary interacting jump particle systems associated with Kac’s simplified model of the Boltzmann equation; we also study the extensions of the techniques they used, which are based on optimal mass transportation arguments, to establish sharp uniform propagation of chaos estimates for the Boltzmann equation for Maxwell molecules. Finally, we extended results and techniques of M. Hairer, focused on the convergence rate of SDEs driven by fractional Brownian motion with additive noise, to a family of multiplicative noises.

Lévy trees.

We proved a Ray-Knight theorem, which provides a continuum genealogical description, for flows of branching processes with competition, in terms the local-times of interactively pruned Lévy-trees.

Optimization in financial markets.

We used entropy minimization techniques to develop a new general convex duality method to solve the utility maximization problem in continuous time financial markets under model uncertainty.

The KPZ universality class.

During recent years we have worked on several variational problems involving the Airy processes, which govern the long time, large scale spatial fluctuations of random growth models in the one dimensional Kardar-Parisi-Zhang universality class. We published a review paper (chapter in the volume Topics in Percolative and Disordered Systems, Springer, 2014), which surveys all their previous results and puts them in the context of the KPZ class. We also studied the asymmetric exclusion process (ASEP) with half-periodic initial condition. We obtained exact formulas for a certain generating function associated with the system. These formulas allowed them to derive formally the conjectured asymptotics for this process, which is the first result of this kind for a process that is not determinantal with this type of initial data.

Editorial work

Some of our responsibilities activities in scientific associations and journals during the period are: S. Martínez and J. San Martín are Members of the Editorial Board of ALEA Latin American Journal of Probability and Mathematical Statistics; A. Maass is member of the International Scientific Committee of the ICSASG consortium to produce a reference genomics sequence for the Atlantic salmon, formed by Canada, Chile and Norway.