Related papers: A Generalized Rough Super Brownian Motion
We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the…
We focus on the existence and characterization of the limit for a certain critical branching random walks in time-space random environment in one dimension which was introduced by M. Birnkenr et.al. Each particle performs simple random walk…
We discuss the compact support property of the rough super-Brownian motion constructed as a scaling limit of a branching random walk in static random environment. The semi-linear equation corresponding to this measure-valued process is the…
We prove that supercritical branching random walk on a transient graph converges almost surely under rescaling to a random measure on the Martin boundary of the graph. Several open problems and conjectures about this limiting measure are…
This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function…
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.…
We focus on the existence and its characterization of limit for a certain critical branching random walks in time-space random environment in 1 dimension which was introduced by Birkner et.al. Each particle performs simple random walk on…
The limiting extremal processes of the branching Brownian motion (BBM), the two-speed BBM, and the branching random walk are known to be randomly shifted decorated Poisson point processes (SDPPP). In the proofs of those results, the Laplace…
We consider branching random walks on the Euclidean lattice in dimensions five and higher. In this non-Markovian setting, we first obtain a relationship between the equilibrium measure and Green's function, in the form of an approximate…
We consider a superprocess with coalescing Brownian spatial motion. We first prove a dual relationship between two systems of coalescing Brownian motions. In consequence we can express the Laplace functionals for the superprocess in terms…
Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex…
Motivated by L\'{e}vy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be…
This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching…
The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity.…
In \cite{SzT}, D. Sz\'asz and A. Telcs have shown that for the diffusively scaled, simple symmetric random walk, weak convergence to the Brownian motion holds even in the case of local impurities if $d \ge 2$. The extension of their result…
Let $\{\mm_n, n=0,1,...\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\mm_n$ normalized by its mean.…
We establish a general sufficient condition for a sequence of Galton Watson branching processes in varying environment to converge weakly. This condition extends previous results by allowing offspring distributions to have infinite…
We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under…
We consider a Branching Random Walk on $\R$ whose step size decreases by a fixed factor, $0<b<1$, with each turn. This process generates a random probability measure on $\R$, that is, the limit of uniform distribution among the $2^n$…
We consider an outward degenerate drifted Brownian motion in the quarter plane with oblique reflections on the boundaries. In this article, we explicitly compute the Laplace transforms of the Green's functions associated with the process.…