This note is devoted to prove that the supremum of a fractional Brownian motion with Hurst parameter $H\in \left( 0,1\right)$ has an infinitely differentiable density on $\left( 0,\infty \right)$. The proof of this result is based on the techniques of…

The asymmetric $k$-step exclusion processes are the simplest interacting particle systems whose hydrodynamic equation may exhibit both increasing and decreasing entropic shocks under Euler scaling. We prove that, under Riemann initial condition with right density zero and adequate left density,…

Sufficient conditions are given for the strict convexity of the limit shape in standard first-passage percolation. These conditions involve (1) asymptotic ``straightness'' of the geodesics, and (2) existence of mean-zero limit distributions for the first-passage times.

We define the first passage bridge from 0 to $\lambda$ as the Brownian motion on the time interval $[0,1]$ conditioned to first hit $\lambda$ at time 1. We show that this process may be related to the Brownian bridge, the…

We disprove a conjecture of Russ Lyons---that for every locally finite, connected graph $G$, the critical probability for (Bernoulli bond) percolation on $G$ is equal to the "first moment method" lower bound on this probability---and propose a possible alternative.

We derive the distribution of the center of mass S of the integrated superBrownian excursion (ISE) from the asymptotic distribution of the Wiener index for simple trees. Equivalently, this is the distribution of the integral of a Brownian snake. A…

We show a method and the structure to calculate the characteristic functions of quadratic Wiener functionals by using classical Weierstrass-Hadamard's theory on entire functions. We also examine the idea by an example for Gaussian processes with multiple Markovian property.

In this note we prove strong invariance principles between ranked excursion lengths and heights of a simple random walk and those of a standard Brownian motion. Some consequences concerning limiting distributions and strong limit theorems will also be presented.

We give a sufficient criterion for the weak disorder regime of directed polymers in random environment, which extends a well-known second moment criterion. We use a stochastic representation of the size-biased law of the partition function.

We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from $Z^2$ by removing all horizontal edges off the $x$-axis, has…