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 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…

Let $(B_t \colon t \ge 0)$ be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=\log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point $x$ in the plane such that ${\mathcal H}^{\phi_\alpha}(\{t \ge 0 \colon…