Related papers: The spans in Brownian motion

Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2 \pi N 1_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly,…

The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of the Brownian motion on $\mathsf{GL}(N;\mathbb{C}),$ in the sense of $\ast $-distributions. The natural candidate for the large-$N$ limit of the empirical distribution…

We consider a super-Brownian motion $X$. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the $\epsilon$-neighborhood for the range of the…

It is well-known (see Dvoretzky, Erd{\H o}s and Kakutani [8] and Le Gall [12]) that a planar Brownian motion $(B_t)_{t\ge 0}$ has points of infinite multiplicity, and these points form a dense set on the range. Our main result is the…

A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$, where $\mathcal{T}$ is the rooted real tree naturally associated with a Brownian excursion and $\phi$ is a random continuous function from $\mathcal{T}$ into…

Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over the whole real line such that B_i(0)=0. We consider the nth iterated Brownian motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of…

We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that there exists a random subset $\Theta$ of $\mathbb{S}^{d-1}$ such that the limit of the derivative martingale exists simultaneously for all directions $\theta \in…

We consider a process given by a two-dimensional fractional Brownian motion with Hurst parameter 1/3 < H < 1/2, along with an associated L\'evy area, and prove the smoothness of a density for this process with respect to Lebesgue measure.

We consider super processes whose spatial motion is the $d$-dimensional Brownian motion and whose branching mechanism $\psi$ is critical or subcritical; such processes are called $\psi$-super Brownian motions. If…

This paper investigates $\pi_T(a_1,a_2) = \mathbb{P}\left(\sup\limits_{t\in[0,T]} (\sigma_1B(t)-c_1t)>a_1, \sup\limits_{t\in[0,T]}( \sigma_2 B(t)-c_2t)>a_2\right),$ where $\{B(t) : t \geq 0\}$ is a standard Brownian motion, with $T >0,…

The signature of Brownian motion in $\mathbb{R}^{d}$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\geq 2$, almost all Brownian…

The signature of a $d$-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over $\RR^{d}$. In this note, we derive the exact rate of…

We study branching Brownian motion in hyperbolic space. As hyperbolic Brownian motion is transient, the normalised empirical measure of branching Brownian motion converges to a random measure $\mu_\infty$ on the boundary. We show that the…

We consider the Brown measure of the free circular Brownian motion, $\boldsymbol{a}+\sqrt{t}\boldsymbol{x}$, with an arbitrary initial condition $\boldsymbol{a}$, i.e. $\boldsymbol{a}$ is a general non-normal operator and $\boldsymbol{x}$…

Brownian motion is a ubiquitous physical phenomenon across the sciences. After its discovery by Brown and intensive study since the first half of the 20th century, many different aspects of Brownian motion and stochastic processes in…

We investigate the sample path properties of Martin-L\"of random Brownian motion. We show (1) that many classical results which are known to hold almost surely hold for every Martin-L\"of random Brownian path, (2) that the effective…

Fine regularity of stochastic processes is usually measured in a local way by local H\"older exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian…

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset \mathbf {R}^d$ are arbitrary nonrandom compact sets. Our…

For $d \geq 2$ let $B$ be standard $d$-dimensional Brownian motion. For any $\alpha < 1/d$ we construct an $\alpha$-H\"{o}lder continuous function $f \colon [0,1] \to \mathbb{R}^d$ so that the range of $B-f$ covers an open set. This…

In a series of recent preprints, we have proven that with probability one the Hausdorff dimension on the outer boundary of planar Brownian motion is 4/3, confirming a conjecture by Mandelbrot. It is also shown that the Hausdorff dimension…