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Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish…
In physics and engineering literature, the distribution of the excursion time of a stationary Gaussian process has been approximated through a method based on a stationary switch process with independently distributed switching times. The…
Area fluctuations of a Brownian excursion are described by the Airy distribution, which found applications in different areas of physics, mathematics and computer science. Here we generalize this distribution to describe the area…
In this note, we study the asymptotical frontier behavior of a branching reflected Brownian motion. There is essentially no difference in maximal displacement between a branching Brownian motion and its reflected counterpart. We provide two…
The purpose of this work is to construct a {\it Brownian motion} with values in simplicial complexes with piecewise differential structure. In order to state and prove the existence of such Brownian motion, we define a family of continuous…
Coalescing simple random walks in the plane form an infinite tree. A natural directed distance on this tree is given by the number of jumps between branches when one is only allowed to move in one direction. The Brownian web distance is the…
Fractional Brownian motion is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically fractional Brownian motion confined to a finite…
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is…
We propose a macroscopic realization of planar Brownian motion by vertically vibrated disks. We perform a systematic statistical analysis of many random trajectories of individual disks. The distribution of increments is shown to be almost…
Upon almost-every realisation of the Brownian continuum random tree (CRT), it is possible to define a canonical diffusion process or `Brownian motion'. The main result of this article establishes that the cover time of the Brownian motion…
In this paper we apply the spectral theory of linear diffusions to study the one-dimensional Liouville Brownian Motion and Liouville Brownian excursions from a given point. As an application we estimate the fractal dimensions of level sets…
We study the distribution of the exit place of iterated Brownian motion in a cone, obtaining information about the chance of the exit place having large magnitude. Along the way, we determine the joint distribution of the exit time and exit…
We derive explicit formulas for probabilities of Brownian motion with jumps crossing linear or piecewise linear boundaries in any finite interval. We then use these formulas to approximate the boundary crossing probabilities for general…
We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations…
Brownian motions in the infinite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent…
There is a close connection between intersections of Brownian motion paths and percolation on trees. Recently, ideas from probability on trees were an important component of the multifractal analysis of Brownian occupation measure, in joint…
We prove that, after suitable rescaling, the simple random walk on the trace of a large critical branching random walk converges to the Brownian motion on the integrated super-Brownian excursion.
In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal…
Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random…
We calculate the distribution of the size of the percolating cluster on a tree in the subcritical, critical and supercritical phase. We do this by exploiting a mapping between continuum trees and Brownian excursions, and arrive at a…