PUBLICATIONS AND PREPRINTS
Behavior dominated by slow particles in a disordered asymmetric exclusion process.
Annals of Applied Probability Vol.14. No.3. 1577 - 1602 (2004) (with I. Grigorescu and T. Seppalainen)Abstract : 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.
Stochastic Processes and their Applications Vol.110. Issue 1. 111 - 143 (2004) (with I. Grigorescu)
Abstract : We consider a system of $N$ Brownian particles evolving independently in a domain $D$. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles. We prove the hydrodynamic limit for the empirical measure process and the average number of visits to the boundary as $N$ approaches infinity.
Journal of Theoretical Probability Vol 16. no 1. 147 - 159 (with I. Grigorescu)
Abstract : 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 coupled 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.
Journal of Theoretical Probability Vol 15. no 3. 817 - 844 (with I. Grigorescu)
Abstract : In an interval containing the origin we study a Brownian motion returning to zero as soon as it reaches the boundary and starting over again, which represents a model for double knock-out barrier options in derivative markets. 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 coupled.
Electronic Communications in Probability Vol.6. (2000) (with R. Atar and S. Athreya)
Abstract : We consider ballistic deposition on a finite strip in a plane for a fixed number of columns. We prove both the upper and the lower bound on the growth rate of the maximal height process of which dependence on the number of columns is also given asymptotically.
Path Collapse for Multidimensional Brownian Motion with Rebirth.
Accepted in Statistics & Probability Letters (with I. Grigorescu)Abstract : In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. For convex or polyhedral regions coupled paths which differ at start by a vector from a set of codimension one collapse with positive probability.
Submitted (with I. Grigorescu)
Abstract : We consider a system of $N$ Brownian particles evolving independently in a domain $D$. As soon as one particle reaches the boundary it is killed and one of the other particles splits into two independent particles. We determine the exact law of the tagged particle as $N$ approaches infinity. In addition, we show that any finite number of labelled particles become independent in the limit.
Ergodic Properties of Multidimensional Brownian Motion with Rebirth.
Submitted (with I. Grigorescu)Abstract : In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Laplacian with pole at the point of rebirth. We determine the resolvent in closed form and prove the exponential ergodicity by Laplace transform methods using the analytic semigroup properties of the Dirichlet Laplacian.
Preprint (with I. Grigorescu)
Abstract : We shall present a general class of deposition models generated by linear operators in the max-plus algebra of the configuration space. The property enables the immediate derivation of a microscopic Hopf-Lax formula which leads to a law of large numbers for the interface under Euler scaling by a method due to T. Seppalainen. Several well known examples are given such as ballistic deposition, the interface growth models associated to totally asymmetric simple exclusion process, totally asymmetric zero range process and totally asymmetric K-exclusion process respectively.
Asymptotic Behavior of Solutions of One Dimensional Parabolic Stochastic Partial Differential Equations. Ph.D. Dissertation
Abstract : We investigate the weak convergence as time goes to infinity of the solutions for parabolic stochastic partial differential equations in one dimension with Dirichlet boundary conditions. We first prove a result for the linear case, using the fact that the solution is Gaussian. Then we get a result for non-linear drift case by proving a comparison theorem in stochastic difference differential equations.