Noise sensitivity of functions on the leaves of a binary tree is studied, and a hypercontractive inequality is obtained. We deduce that the spider walk is not noise stable.

Recently, Kalvin M. Jansons derived in an elegant way the Laplace transform of the time spent by an excursion above a given level $a>0$. This result can also be derived from previous work of the author on the occupation time…

We present a Gaussian correlation inequality which is closely related to a result of Schechtman, Schlumprecht and Zinn (1998) on the well-known Gaussian correlation conjecture. The usefulness of the inequality is demonstrated by several important applications to the estimates of…

Consider two exchangeable sequences $(X_k)_{k \in N}$ and $(\hat{X}_k)_{k \in N}$ with the property that $S_n \equiv \sum_{k=1}^n X_k$ and $\hat{S}_n \equiv \sum_{k=1}^n \hat{X}_k$ have the same distribution for all $n \in N$. David Aldous posed the following question. Does…

Rogers-Pitman have shown that the sum of the absolute value of $B^{(\mu)}$, Brownian motion with constant drift $\mu$, and its local time $L^{(\mu)}$ is a diffusion $R^{(\mu)}$. We exploit the intertwining relation between $B^{(\mu)}$ and $R^{(\mu)}$ to show that the…

The purpose of this note is to make available a reasonably complete and straightforward proof of Strassen's theorem on stochastic domination, and to draw attention to the original paper. We also point out that the maximal possible value of $P(Z…

Fleming-Viot processes incorporating mutation and selection are considered. It is well-known that if the mutation factor is of uniform type, the process has a reversible stationary distribution, and it has been an open problem to characterize the class of the…

We consider an isotropic Brownian flow on $R^d$ for $d\geq 2$ with a positive Lyapunov exponent, and show that any nontrivial connected set almost surely contains points whose distance from the origin under the flow grows linearly with time. The…

Let $T_1$ denote the first passage time to 1 of a standard Brownian motion. It is well known that as $\lambda$ goes to infinity, $P\{ T_1 > \lambda \}$ goes to zero at rate $c \lambda^{-1/2}$, where $c$ equals $(2/…

We prove that an optional process of non-exploding realized power variation along stopping times possesses almost surely làglàd paths. This result is useful for the analysis of some imperfect market models in mathematical finance. In the finance applications variation naturally…