Kasun Fernando

During the past few decades there has been a great enthusiasm among the dynamicists to study statistical properties of dynamical systems. This enthusiasm is not superfluous or artificial in that it is based on the physical imperative to predict the long term behaviour of these systems and determine how accurate these predictions are. This is exactly the central theme of Ergodic Theory -- as opposed to looking at individual points (particles) in a system, looking at them collectively and determining the asymptotics of the system as a whole. For example, it is impossible to keep track of the velocities (kinetic energy) of individual gas particles in a gas chamber. However, by observing the temperature, pressure and etc. of the chamber over time and using the dynamics of the physical model of the system, one can predict (preferably up to a controlled error) the future behaviour of the system.

To be more precise, dynamicists are interested in obtaining results like the Central Limit Theorem, Local Limit Theorems, Almost Sure Invariance Principles and Large Deviation Principles in the setting of deterministic dynamical systems. It is well-known that these are satisfied by independent and identically distributed random variables under appropriate moment conditions. Therefore, it is reasonable to believe that systems that satisfy these are asymptotically well-behaved at a macroscopic level. It is also generally understood that the more chaotic the systems are the better statistical properties they would exhibit. This is more or less evident from the fact that most of the results obtained in this regard are for systems that have some hyperbolicity. The heuristic is that hyperbolicity allows sufficient amount of mixing in the system so that observations become more and more uncorrelated while the system evolves over time.

Whenever the asymptotic behaviour of a system is studied, it is meaningful to control the error of approximation by finite time observations. An analogue of this in the abstract setting of the Central Limit Theorem for independent identically distributed random variables is the use a uniform asymptotic expansion called the Edgeworth Expansion to describe the error in approximating the finite distributions by the normal distribution. Also, there are extensions of this theory of asymptotic expansions to independent random variables which are not necessarily identically distributed. However, this is inadequate in the setting of dynamical systems because the observations are far from being independent (their deterministic nature implies that the immediate past determines the present). My current research is primarily focused on possible extensions of this theory of asymptotic expansions to more general settings that are not included in the classical theory, including the case of random variables arising as observations of chaotic dynamical systems.

Projects

  1. Extension of the classical theory of Edgeworth expansions to non-lattice discrete random variables. Joint with Dmitry Dolgopyat. (submitted)
  2. Asymptotic expansions for the central limit theorem of weakly dependent random variables with applications to dynamical systems. Joint with Carlangelo Liverani. (in preparation)
  3. Asymptotic Expansions for Large Deviations with applications to stochastic differential equations. Joint with Pratima Hebbar. (ongoing)
  4. Limit Theorems for cocycles and related asymptotic expansions with applications to the KZ cocylcle. Joint with Hamid Al-Saqban. (ongoing) 

More detailed descriptions about these and future projects can be found in my research statement.