## 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.

- Y. Bakthin, Zs. Pajor-Gyulai
*Metastability and cycle structure in strictly attracting noisy heteroclinic networks*In preparation. - Y. Bakhtin, Zs. Pajor-Gyulai
*Rare transitions in noisy planar heteroclinic networks*In preparation. - Y. Bakhtin, Zs. Pajor-Gyulai
*Malliavin calculus approach to long exits from a neighborhood of an unstable equilibrium on the line.*Submitted to Annals of Applied Probability - Y. Bakhtin, Zs. Pajor-Gyulai
*Scaling limit for escapes from unstable equilibria in the vanishing noise limit: nontrivial Jordan block case*Submitted to Stochastics and Dynamics

*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.

- M. Hairer, G. Iyer, L.Koralov, A. Novikov, Zs. Pajor-Gyulai
*A fractional kinetic process describing the intermediate time behaviour of cellular flows.*To appear in the Annals of Probability - M. Hairer, L. Koralov, Zs. Pajor-Gyulai
*From averaging to homogenization in cellular flows - an exact description of the transition*Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. Vol. 52. No. 4., 2016

*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.

- Zs. Pajor-Gyulai, M. Salins
*On dynamical systems perturbed by a null-recurrent fast motion: The continuous coefficient case with independent driving noises*Journal of Theoretical Probability 2015, p1-17. - Zs. Pajor-Gyulai, M. Salins
*On dynamical systems perturbed by a null-recurrent motion: The general case*Stochastic Processes and their Applications Vol. 127. No.6 p1960-1997, 2017

#### 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.

#### Random walk model for the two disk planar Lorentz process

In this project, I investigated how to derive a toy model for the two disk planar Lorentz process, i.e. the two interacting hard disks suffering specular (this is when the angle of incidence equals the angle of reflection) collisions in a fixed periodic configuration of scatterers and exchanging energy upon meeting.

- Zs. Pajor-Gyulai, D.Szász
*Weak convergence of Random Walk Conditioned to Stay Away from Small Sets*Studia Scientiarum Mathematicarum Hungarica Vol. 50 No. 1, p122-128, 2013 - Zs. Pajor-Gyulai, D. Szász
*Perturbation approach to scaled type Markov renewal processes with infinite mean*Preprint, 2012 - Zs. Pajor-Gyulai, D.Szász
*Energy Transfer and Joint Diffusion*Journal of Statistical Physics Vol. 146, No. 5, p1001-1025, 2012 - Zs. Pajor-Gyulai, D.Szász, I.P.Toth
*Billiard models and energy transfer*in XVIth International Congress on Mathematical Physics, p328-332, World Sci. Publ., Hackensack, NJ, 2012