相关论文: The height gap of planar Brownian motion is $\frac…

In this article we study the convex hull spanned by the union of trajectories of a standard planar Brownian motion, and an independent standard planar Brownian bridge. We find exact values of the expectation of perimeter and area of such a…

In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are different and mainly…

We establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected values of the perimeter and the area of the Brownian convex…

We present an exact solution for the probability density function $P(\tau=t_{\min}-t_{\max}|T)$ of the time-difference between the minimum and the maximum of a one-dimensional Brownian motion of duration $T$. We then generalise our results…

The conditional density of Brownian motion is considered given the max, B(t|\max), as well as those with additional information: B(t|close, max), B(t|close, max, min) and B(t|max, min) where the close is the final value: B(t=1)=c and t in…

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…

We provide upper and lower bounds for the mean ${\mathscr M}(H)$ of $\sup_{t\geqslant 0} \{B_H(t) - t\}$, with $B_H(\cdot)$ a zero-mean, variance-normalized version of fractional Brownian motion with Hurst parameter $H\in(0,1)$. We find…

Known results show that the diameter $d_1$ of the trace of planar Brownian motion run for unit time satisfies $1.595 \leq \mathbb{E} d_1 \leq 2.507$. This note improves these bounds to $1.601 \leq \mathbb{E} d_1 \leq 2.355$. Simulations…

We present a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. We include a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian…

We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian…

We construct Brownian motion on a wide class of metric spaces similar to graphs, and show that its cover time admits an upper bound depending only on the length of the space.

Let B_t be a planar Brownian loop of time duration 1 (a Brownian motion conditioned so that B_0 = B_1). We consider the compact hull obtained by filling in all the holes, i.e. the complement of the unique unbounded component of R^2\B[0,1].…

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau=t_{\min}-t_{\max}$ between the time $t_{\min}$ of the global…

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 study the convex hull of planar Brownian motion run until the exit time from the unit disk. Our primary objective is to compute the expected perimeter of this convex hull, thereby complementing recent results on the convex hull of…

A uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains is established. Exact Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are given. Extensions…

We derive a new theoretical lower bound for the expected supremum of drifted fractional Brownian motion with Hurst index $H\in(0,1)$ over (in)finite time horizon. Extensive simulation experiments indicate that our lower bound outperforms…

Based on an optimal rate wavelet series representation, we derive a local modulus of continuity result with a refined almost sure upper bound for fractional Brownian motion. \sloppy The obtained upper bound of the small fractional Brownian…

This paper describes joint work with Oded Schramm and Wendelin Werner establishing the values of the planar Brownian intersection exponents from which one derives the Hausdorff dimension of certain exceptional sets of planar Brownian…

Consider a planar Brownian motion run for finite time. The frontier or ``outer boundary'' of the path is the boundary of the unbounded component of the complement. Burdzy (1989) showed that the frontier has infinite length. We improve this…