Zsolt Pajor-Gyulai, Courant Instructor

Research

Throughout my research career, I mostly studied asymptotic problems arising from random perturbations of continuous time dynamical systems. It is a fundamental problem in mathematics, with applications to physical and life sciences, to take the underlying complex and usually incomplete microscopic dynamics of a certain system and derive the macroscopic behavior. Such systems often posess multiple spatial and time scales. When they are very large or observed for a long time, asymptotic methods offer a mathematical framework to explain how to average small contributions of individual elements in the microscopic picture to obtain information about the macroscopic behavior of the system.

Ergodic properties of noisy heteroclinic networks and behavior around equilibria

Currently, I am studying the long time behavior of a diffusion along a heteroclinic network, which is collections of hyperbolic saddle points and heteroclinic orbits connecting them. One such example is a noisy cellular flow (see below), however, heteroclinic networks are common in different contexts ranging from population dynamics to the modelling of neural processes.

Rare paths along the network: The first hyperbolic equilibria repells the particles and therefore the overwhelming majority of them follow the U-shaped trajectory. On the other hand, certain slower particles (their fraction scales polynomially in the size of the noise) withstand the repulsion and thus are able to exit through the left and the bottom.

Anomalous diffusion and fractional kinetic in planar cellular flows

In this project, we studied the the macroscopic transport properties of periodic, incompressible, planar cellular flows on intermediate time scales. We established a fractional kinetic effective process whose variance grows as the square root of the elapsed time. This grows turns smoothly into the homogenized linear one for time scales that grow faster than the inverse of the noise intensity.

Particles starting on the separatrix between the cells. The empirical variance of the particle cloud grows proportionally to the square root of the elapsed time. On very long timescales, there is a smooth transition to the homogenization regime where the variance grows linearly.

Dynamical systems with perturbation driven by a null recurrent fast motion

In this project, we studied a multidimensional diffusion process with a slow and a fast component where the fast motion is null-recurrent and the slow motion only deviates significantly from some deterministic dynamics when the fast one is in some compact set.

Critical behavoir of random polymers

In this project, we investigated a two-parameter asymptotic problem on the behavior of a continuous random three (or higher) dimensional homopolymer in an attracting potential. We studied the situation when the length of the polymer tends to infinity, and the temperature simultaneously approaches the critical value at which a phase transition occurs between a densely packed globular state and an exteded phase.