Probability Theory and Related Fields
Abstract :
We study the existence and asymptotic properties of a conservative branching particle system driven by a diffusion with smooth coefficients for which birth and death are triggered by contact with a set.
Sufficient conditions for the process to be non-explosive are given. In the Brownian
motions case the domain of evolution can be non-smooth, including Lipschitz, with
integrable Martin kernel. The results are valid for an arbitrary number of particles
and non-uniform redistribution after branching. Additionally, with probability one, it is shown
that only one ancestry line survives. In special cases, the evolution of the surviving particle is
studied and for a two particle system on a half line we derive explicitly the transition function
of a chain representing the position at successive branching times.
steady state and scaling limit for a traffic congestion model
European Series in Applied and Industrial Mathematics ESAIM: PS 14 pp. 271-285
Abstract :
In a general model (AIMD) of transmission control protocol (TCP) used in internet traffic congestion
management, the time dependent data flow vector x_{i}(t) > 0 undergoes a biased random walk on two distinct scales. The amount of data of each component x(t) goes up to x_{i}(t) + a with probability 1 –  p_{i}(x) on a unit scale or down to c x_{i}(t), 0 < c < 1 with probability P_{i}(x). on a logarithmic scale, where ?i depends on the joint state of the system x.  We investigate the long time behavior, mean field limit, and the one particle case. According to c = lim inf|x|?° |x|?i(x), the process drifts to ° in the subcritical c < c+(n,?) case and has an invariant probability measure in the supercritical case c > c+(n,?). Additionally, a scaling limit is proved when ?i(x) and are of order N?1 and t ? Nt, in the form of a
continuum model with jump rate ?(x).
ergodic properties of multidimensional brownian motion with rebirth
Electron. J. Probab. 12, no. 48, pp. 1299-1322
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 Dirichlet Laplacian centered at the point of return. We determine the re- solvent in closed form and prove the exponential ergodicity by Laplace transform methods using the analytic semigroup properties of the Dirichlet Laplacian. In d = 1
we calculate the exact spectrum of the process and note that the principal eigenvalue is equal to the second eigenvalue of the Dirichlet Laplacian, correcting an error from a previous paper. The method can be generalized to other renewal type processes.
behavior dominated by slow particles in a disordered asymmetric exclusion process
Annals of Applied Probability Vol.14. No.3. pp. 1577 – 1602
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.
hydrodynamic limit for a fleming-viot system
Stochastic Processes and their Applications Vol.110. Issue 1. pp. 111 – 143
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.
path collapse for an inhomogeneous random walk
Journal of Theoretical Probability Vol 16. no 1. pp. 147 – 159
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 modeled 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.
brownian motion on the figure eight
Journal of Theoretical Probability Vol 15. no 3. pp. 817 – 844
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.
ballistic deposition on a planar strip
Electronic Communications in Probability Vol.6.
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
Statistics & Probability Letters Vol. 70. no. 3. pp. 199 – 209
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 co-dimension one collapse with positive probability.
tagged particle limit for a fleming-viot type system
Electronic Journal of Probability, Vol.11.
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.
the doeblin condition for a class of diffusions with jumps
Abstract :
We prove non-explosiveness and a lower bound of the spectral gap via the strong Doeblin condition for a large class of stochastic processes evolving in the interior of an open set D Rd according to an underlying Markov process with transition probabilities p(t; x; dy), undergoing jumps to a random point x in D with distribution (dx) as soon as they reach a point outside D. Besides usual regularity conditions on p(t; x; dy), we give conditions on the family of measures preventing mass from escaping
to the boundary in the long run, even when the family is not tight. The setup can be applied to a multitude of models considered recently, including a particle system with the Bak-Sneppen dynamics from evolutionary biology.
fixation time for an evolution model
to appear in Stochastic Models
Abstract :
We study the asymptotic value as L ? ° of the time for evolution ?, un- derstood as
the first time to reach a preferred word of length L using an alphabet with N letters. The
word is updated at unit time intervals randomly but configurations with letters matching with the
preferred word are sticky, i.e. the probability to leave the con- figuration equals 0 ² ? ² 1, where ?
may depend on the configuration. The model is introduced in [1] in the case ? = 0, where it
was shown that E[?] ? N ln(L). We first give a simple proof that the mode of ? has the same
asymptotic value. When ? ?= 0, ? is exponential. We further show that the natural scaling is ? = O(L?1) and study the interacting case when ? depends linearly on the number of matches with the preferred
word when the scaling limit of the number of non-matching letters follows a Galton-Watson
process with immigration. Under similar scaling, the empirical measure converges to the solution
of a discrete logistic equation.
markov processes with redistribution
to appear in Markov Processes and Related Fields
Abstract :
We study a class of stochastic processes evolving in the interior of a set D
according to an underlying Markov kernel, undergoing jumps to a random point x in D
with distribution (dx) as soon as they reach a point outside D. We give conditions on
the family of measures (dx) preventing that inFInitely many jumps occur in nite time
(explosiveness), conditions for ergodicity and the existence of a the spectral gap. The
setup is applied to a multitude of models considered recently, including particle systems
like the Fleming-Viot branching process and a new variant of the Bak-Sneppen dynamics
from evolutionary biology. The last part of the paper is expository and discusses the
relation with quasi-stationary distributions.


hydrodynamics for max-plus linear interface growth models
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.