Wael Bahsoun

Rigorous numerical approximation of escape rates

An interval map with holes is a mathematical model which is used in the study of nonequilibrium statistical mechanics. We use Ulams method to approximate the escape rate for an interval map with holes and find a bound on the approximation error.

METASTABILITY OF CERTAIN INTERMITTENT MAPS

We study an intermittent map which has exactly two ergodic invariant densities. The densities are supported on two subintervals with a common boundary point. Due to certain perturbations, leakage of mass through subsets, called holes, of the initially invariant subintervals occurs and forces the subsystems to merge into one system that has exactly one invariant density. We prove that the invariant density of the perturbed system converges in the L1-norm to a particular convex combination of the invariant densities of the intermittent map. In particular, we show that the ratio of the weights in the combination is equal to the limit of the ratio of the measures of the holes.

Decay of correlation for random intermittent maps

We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate.

Mixing rates and limit theorems for random intermittent maps

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps

Markov Switching for Position Dependent Random Maps with Application to Forecasting

T_\alpha

Absolutely Continuous Invariant Measures that Cannot be Observed Experimentally

using the full parameter range

Weakly Convex and Concave Random Maps with Position Dependent Probabilities

0< \alpha < \infty

Almost sure Nash equilibrium strategies in evolutionary models of asset markets

, in general. We derive a number of results around a common theme that illustrates in detail how the constituent map that is fastest mixing (i.e.\ smallest

STOCHASTIC PERTURBATIONS OF POSITION DEPENDENT RANDOM MAPS

\alpha

Ergodic Theory, Open Dynamics, and Coherent Structures

) combined with details of the randomizing process, determines the asymptotic properties of the random transformation. Our key result (Theorem 1.1) establishes sharp estimates on the position of return time intervals for the \emph{quenched} dynamics. The main applications of this estimate are to \textit{limit laws} (in particular, CLT and stable laws, depending on the parameters chosen in the range