We consider invasion percolation on the square lattice. van den Berg, Peres, Sidoravicius and Vares have proved that the probability that the radius of a so-called pond is larger than n, differs at most a factor of order log n…

Let $X$ be a Banach space. Suppose that for all $p\in (1, \infty)$ a constant $C_{p,X}$ depending only on $X$ and $p$ exists such that for any two $X$-valued martingales $f$ and $g$ with tangent martingale difference sequences one has…

A one dimensional stochastic differential equation of the form \[dX=A X dt+\tfrac12 (-A)^{-\alpha}\partial_\xi[((-A)^{-\alpha}X)^2]dt+\partial_\xi dW(t),\qquad X(0)=x\] is considered, where $A=\tfrac12 \partial^2_\xi$. The equation is equipped with periodic boundary conditions. When $\alpha=0$ this equation arises in the Kardar-Parisi-Zhang model. For $\alpha\ne 0$,…

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance $$ \int^{s\wedge t}_0 u^a [(t-u)^b+(s-u)^b]du, $$ parameters $a>-1$, $-1 < b\leq 1$, $|b|\leq 1+a$, corresponds to fractional Brownian motion for $a=0$, $-1 < b…