Related papers: On Conditioning Brownian Particles to Coalesce
In a series of essays, beginning with this article, we are going to develop a new formulation of micro-phenomena based on the principles of reality and causality. The new theory provides with us a new depiction of micro-phenomena assuming…
Consider a massive (inert) particle impinged from above by N Brownian particles that are instantaneously reflected upon collision with the inert particle. The velocity of the inert particle increases due to the influence of an external…
Given a random time, we characterize the set of martingales for which the stopping theorems still hold. We also investigate how the stopping theorems are modified when we consider arbitrary random times. To this end, we introduce some…
We introduce a natural family of random walks on the set of integers that scale to fractional Brownian motion. The increments X_n have the property that given {X_k: k < n}, the conditional law of X_n is that of X_{n-k_n}, where k_n is…
In this thesis, we study asymptotic properties of the standard branching Brownian motion, with a specific emphasis on the additive martingales at high temperature. We start by presenting classic and fundamental tools for our investigation.…
Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion…
Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in a work of Bertoin, where special attention was…
At high temperature, the overlap of two particles chosen independently according to the Gibbs measure of the branching Brownian motion converges to zero as time goes to infinity. We investigate the precise decay rate of the probability to…
We study the zero temperature limit for interacting Brownian particles in one dimension with a pairwise potential which is of finite range and attains a unique minimum when the distance of two particles becomes a>0. We say a chain is formed…
The Brownian Web (BW) is a family of coalescing Brownian motions starting from every point in space and time $\R\times\R$. It was first introduced by Arratia, and later analyzed in detail by T\'{o}th and Werner. More recently, Fontes,…
The weak limits of the measure-valued processes organized as a mass carried by the interacting Brownian particles are described. As a limiting flow the Arrattia flow is obtained.
The notion of fractality, in the context of positive-valued probability distributions, is conventionally associated with the class of Paretian probability laws. In this research we show that the Paretian class is merely one out of six…
It is shown that time reversibility of Hamiltonian microscopic dynamics and Gibbs canonical statistical ensemble of initial conditions for it together produce an exact virial expansion for probability distribution of path of molecular…
A particle subject to successive, random displacements is said to execute a random walk (in position or some other coordinate). The mathematical properties of random walks have been very thoroughly investigated, and the model is used in…
In this paper, we prove a result of equivalence in law between a diffusion conditioned with respect to partial observations and an auxiliary process. By partial observations we mean coordinates (or linear transformation) of the process at a…
We analyze here different types of fractional differential equations, under the assumption that their fractional order $\nu \in (0,1] $ is random\ with probability density $n(\nu).$ We start by considering the fractional extension of the…
We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and…
Particle approximations for certain nonlinear and nonlocal reaction-diffusion equations are studied using a system of Brownian motions with killing. The system is described by a collection of i.i.d. Brownian particles where each particle is…
We derive the asymptotic winding law of a Brownian particle in the plane subjected to a tangential drift due to a point vortex. For winding around a point, the normalized winding angle converges to an inverse Gamma distribution. For winding…
In this paper we study the maximal position process of branching Brownian motion in random spatial environment. The random environment is given by a process $\xi = \left(\xi(x)\right)_{x\in\mathbb{R}}$ satisfying certain conditions. We show…