I am interested in studying Interacting Particle Systems described by systems of stochastic differential equations with singular drifts. These mathematical models are used to describe systems of particles where the behavior of each particle is influenced by the presence and movement of others. Such systems are widely studied in Probability Theory, Statistical Mechanics, and Mathematical Physics. I am particularly interested in constructing these systems in singular cases, studying their trajectory properties and limiting behaviors.
Archimedes' principle, discovered around 256 BC, states that the upward buoyant force exerted on a body immersed in a fluid (or gas) is equal to the weight of the fluid (or gas) that the body displaces. It is astonishing that until modern times, there was no rigorous evidence to prove it based on a molecular model. This type of reasoning fits into the ideas of Hilbert's 6th problem, and I am very interested in particle models that allow us to deduce macroscopic phenomena from simple microscopic interactions.
Recently, I have developed an interest in artificial intelligence in its broadest sense, from both practical and theoretical perspectives, with a particular focus on the application of interacting particle models in neural networks and machine learning.
One of my interests is related to Lévy processes on the real line. More precisely, I work on determining the relationships between various characteristics of one-dimensional Lévy processes, such as Lévy measures, transition probability densities, process supremum and infimum densities, and entrance laws. These relationships further enable the study of the behavior of these processes for limiting parameter values.
Hyperbolic spaces are examples of non-Euclidean geometric spaces characterized by constant negative curvature. My research interests in this area are focused on the potential theory of hyperbolic Brownian motions and related processes, such as geometric Brownian motion, and include the derivation of Poisson kernels and Green's functions for basic bounded and unbounded hyperbolic domains.