We slightly improve the upper bounds of disconnection exponents for planar Brownian motion that we derived in an earlier paper. We also give a proof of the lower bound $1/(2\pi)$ for the disconnection exponent for one path.

Let $B$ be a two dimensional Brownian motion and let the frontier of $B[0,1]$ be defined as the set of all points in $B[0,1]$ that are in the closure of the unbounded connected component of its complement. We prove that…

Using Monte-Carlo simulations, we estimate numerically disconnection exponents for planar Brownian motions. These simulations tend to confirm conjectures by Duplantier and Mandelbrot.

We give another proof of the following result from a joint paper with Bálint Tóth: A Brownian motion reflected on an independent time-reversed Brownian motion is a Brownian motion.

We give a simple proof that in a Lipschitz domain in two dimensions with Lipschitz constant one, there is pathwise uniqueness for the Skorokhod equation governing reflecting Brownian motion.

For the one-dimensional Brownian motion $B=(B_t)_{t\geq 0}$, started at $x>0$, and the  first hitting time $\tau=\inf\{t\geq 0:B_t=0\}$, we find the probability density of $B_{u\tau}$ for a $u\in(0,1)$, i.e. of the Brownian motion on its way to hitting zero.

We present functional versions of recent results on the univariate distributions of the process $V_{x,u} = x + W_{u\tau(x)},$ $0\le u\le 1$, where $W_\bullet$ is the standard Brownian motion process, $x>0$ and $\tau (x) =\inf\{t>0 :\, W_{t}=-x\}$.

We derive integral formulas, involving the Airy function, for moments of the time a two-sided Brownian motion with parabolic drift attains its maximum.

In the present paper we deduce explicit formulas for the probability laws of the quotients $X_t/R_t$ and $m_t/R_t$, where $X_t$ is the standard Brownian motion and $m_t$, $M_t$, $R_t$ are its running minimum, maximum and range, respectively.The computation makes use of standard techniques from…

Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least…