Advances in Stochastic Duality for Interacting Particle Systems: from many to few

Interacting particle systems (IPS) is a subfield of probability theory that provided a fruitful framework in which several questions of physical interests have been answered with mathematical rigor. An interacting particle system is a stochastic system consisting of a very large number of particles interacting with each other. The class of IPS...

Urn models and other approaches to risk and tails, with applications in risk management and climatology

This dissertation collects three scientific contributions, already published in international peer-reviewed journals, plus some extra considerations and work-in-progress. First, we present a model based on reinforced urn processes, which conjugates to the right-censored recovery process, and empirically apply it to the time series of recovery...

The Symmetric Exclusion Process and the Gausian Free Field on compact Riemannian manifolds

In this thesis we study the Symmetric Exclusion Process (SEP) and the Discrete Gaussian Free Field (DGFF) on compact Riemannian manifolds. In particular, we obtain the hydrodynamic limit and the equilibrium fluctuations of SEP and we show that the DGFF converges to its continuous counterpart. To define these discrete models, we construct grids...

Fluctuations for Interacting Particle Systems with Duality

This thesis is concerned with fluctuations of interacting particle systems that<br/>enjoy the property of duality. The main contributions of this work are divided<br/>in two main parts. In the first part we study some of the advantages of looking<br/>at the density fluctuation field through the lenses of orthogonal self-dualities. In<br/>the...

Large deviations for stochastic processes on Riemannian manifolds

This thesis is concerned with large deviations for processes in Riemannian manifolds. In particular, we study the extensions of large deviations for random walks and Brownian motion to the geometric setting. Furthermore, we also consider large deviations for random walks in Lie groups. The additional group structure allows for improvements over...

Gaps, Frequencies and Spacial Limits of Continued Fraction Expansions

In this thesis continued fractions are studied in three directions: semi-regular continued fractions, Nakada’s α-expansions and N-expansions. In Chapter 1 the general concept of a continued fraction is given, involving an operator that yields the partial quotients or digits of a continued fraction expansion. The approximation coefficients θ_n(x)...

Symmetric interacting particle systems: Self-duality and hydrodynamics in dynamic random environment

In this thesis, we study scaling and detailed properties of a class of conservative interacting particle systems. In particular, in the first part we derive the hydrodynamic equation for the symmetric exclusion process in presence of dynamic random environment. The second part of the thesis focuses on a detailed property of conservative particle...

Semigroup methods for large deviations of Markov processes

Coupling, Concentration and Random Walks in Dynamic Random Environments

In this thesis we discuss concentration inequalities, relaxation to equilibrium of stochastic dynamics, and random walks in dynamic random environments. In stochastic systems one is interested in macroscopic and/or asymptotic properties as well as in fluctuations around typical behaviour. But the dependence structure induced by the interaction...