In this paper a nonautonomous dynamical system is considered, a stochastic one that is obtained from the dissipative Euler equation subject to a stochastic perturbation, an additive noise. Absorbing sets have been defined as sets that depend on time and…

Let $X$ be a process defined on an optional random set. The paper develops two different conditions on $X$ guaranteeing that it is the restriction of a uniformly integrable martingale. In each case, it is supposed that $X$ is the…

Given an arbitrary distribution on a countable set, consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent…

Let $M_\alpha$ be the closure of the range of a stable subordinator of index $\alpha\in ]0,1[$. There are two natural constructions of the $M_{\alpha}$'s simultaneously for all $\alpha\in ]0,1[$, so that $M_{\alpha}\subseteq M_{\beta}$ for $0< \alpha < \beta < 1$:…

Orey and Taylor (1974) introduced sets of ``fast points'' where Brownian increments are exceptionally large, ${\rm F}(\lambda):=\{ t\in[0,1]: \limsup_{h\to 0}{ | X(t+h)-X(t)| / \sqrt{ 2h|\log h|}} \ge \lambda\}$. They proved that for $\lambda \in (0,1]$, the Hausdorff dimension of ${\rm…

More thorough results than in our previous paper in Nagoya Math. J. are given on the $L_p$-operator norm estimates for the Kac operator $e^{-tV/2} e^{-tH_0} e^{-tV/2}$ compared with the Schrödinger semigroup $e^{-t(H_0+V)}$. The Schrödinger operators $H_0+V$ to be treated in…

We consider the random walk on supercritical percolation clusters in $\mathbb{Z}^d$. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit…

We show that quaternionic Gaussian random variables satisfy a generalization of the Wick formula for computing the expected value of products in terms of a family of graphical enumeration problems. When applied to the quaternionic Wigner and Wishart families of…

For a much-studied model of random copolymer at a selective interface we prove that the slope of the critical curve in the weak-disorder limit is strictly smaller than 1, which is the value given by the annealed inequality. The proof…

We prove that the Itô map, that is the map that gives the solution of a differential equation controlled by a rough path of finite $p$-variation with $p\in [2,3)$ is locally Lipschitz continuous in all its arguments and we give…