Ilie Grigorescu

Brownian motion on the figure eight

In an interval containing the origin we study a Brownian motion which returns to zero as soon as it reaches the boundary. We determine explicitly its transition probability, prove it is ergodic and calculate the decay rate to equilibrium. It is shown that the process solves the martingale problem for certain asymmetric boundary conditions and can be regarded as a diffusion on an eight shaped domain. In the case the origin is situated at a rationally commensurable distance from the two endpoints of the interval we give the complete characterization of the possibility of collapse of distinct paths.

Behavior dominated by slow particles in a disordered asymmetric exclusion process

We study the large space and time scale behavior of a totally asymmetric, nearest-neighbor exclusion process in one dimension with random jump rates attached to the particles. When slow particles are sufficiently rare the system has a phase transition. At low densities there are no equilibrium distributions, and on the hydrodynamic scale the initial profile is transported rigidly. We elaborate this situation further by finding the correct order of the correction from the hydrodynamic limit, together with distributional bounds averaged over the disorder. We consider two settings, a macroscopically constant low density profile and the outflow from a large jam.

Path Collapse for an Inhomogeneous Random Walk

On an open interval we follow the paths of a Brownian motion which returns to a fixed point as soon as it reaches the boundary and restarts afresh indefinitely. We determine that two paths starting at different points either cannot collapse or they do so almost surely. The problem can be modelled as a spatially inhomogeneous random walk on a group and contrasts sharply with the higher dimensional case in that if two paths may collapse they do so almost surely.

Critical scale for a continuous AIMD model

A scaled version of the general AIMD model of transmission control protocol (TCP) used in Internet traffic congestion management leads to a Markov process x(t) representing the time dependent data flow that moves forward with constant speed on the positive axis and jumps backward to γx(t), 0 < γ < 1 according to a Poisson clock whose rate α(x) depends on the interval swept in between jumps. We give sharp conditions for Harris recurrence and analyze the convergence to equilibrium on multiple scales (polynomial, fractional exponential, exponential) identifying the critical case xα(x) ∼ β. Criticality has different behavior according to whether it occurs at the origin or infinity. In each case, we determine the transient (possibly explosive), null—and positive—recurrent regimes by comparing β to ( − ln γ)− 1.

Phase Transition for Infinite Systems of Spiking Neurons

We prove the existence of a phase transition for a stochastic model of interacting neurons. The spiking activity of each neuron is represented by a point process having rate 1 whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all presynaptic neurons since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to 0 and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effect leading to an abrupt loss of potential occurring at random times driven by an independent Poisson point process of rate

Optimal stopping for Shepp's urn with risk aversion

\gamma > 0 .

Maximizing the discounted survival probability in Vardi's casino

γ>0. For this process we prove the existence of a value