We prove a Chung's law of the iterated logarithm for recurrent linear Markov processes. In order to attain this level of generality, our normalization is random. In particular, when the Markov process in question is a diffusion, we obtain the…

The question considered in this paper is which sequences of $p$-integrable random variables can be represented as conditional expectations of a fixed random variable with respect to a given sequence of sigma-fields. For finite families of sigma-fields, explicit inequality equivalent…

We study the metastability of the stochastic three dimensional Ising model on a finite torus under a small positive magnetic field at very low temperatures.

We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions. The present counterexample only works for first-passage percolation on…

We prove that the Moderate Deviation Principle (MDP) holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, converges in probability at an exponential rate. A consequence of this…

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.

This work was motivated by the recent work by H. Dette, J. Pitman and W.J. Studden on a new duality relation for random walks. In this note we consider the diffusion process limit of their result, and use the alternative…

Let $B$ be a Brownian motion in $R^d$, $d=2,3$. A time $t\in [0,1]$ is called a cut time for $B[0,1]$ if $B[0,t) \cap B(t,1] = \emptyset$. We show that the Hausdorff dimension of the set of cut times equals $1…