Regular g-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories…

Let $\Lambda$ be the limiting smallest eigenvalue in the general $(\beta,a)$-Laguerre ensemble of random matrix theory. That is, $\Lambda$ is the $n\to\infty$ distributional limit of the (scaled) minimal point drawn from the density proportional to $\Pi_1\leq i\leq j\leq n$ $$\left|\lambda_i-\lambda_j\right|^\beta\prod_{i=1}^n\lambda_i^{\frac{\beta}{2}(a+1)-1}e^{-\frac{\beta}{2}\lambda_i}$$…

This note is motivated by results of Angel, Goodman, den Hollander and Slade (2008) and Damron, Sapozhnikov and Vagvolgyi (2009) about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on…

Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random environment (BDRE) conditioned on eventual extinction of the population…

We consider the voter model on $\mathbb{Z}$, starting with all 1's to the left of the origin and all $0$'s to the right of the origin. It is known that if the associated random walk kernel $p$ has zero mean…

Let $X$, $Y$ be continuous-time martingales taking values in a se\-pa\-rable Hilbert space $\mathcal{H}$.(i) Assume that $X$, $Y$ satisfy the condition $[X,X]_t\geq [Y,Y]_t$ for all $t\geq 0$. We prove the sharp inequalities $$ \sup_t||Y_t||_p\leq (p-1)^{-1}\sup_t||X_t||_p,\qquad 1 < p\leq 2,$$ $$…

Let $Y$ be a nonnegative random variable with mean $\mu$ and finite positive variance $\sigma^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by $$ E[Yf(Y)]=\mu E…

Branching annihilating random walk (BARW) is a generic term for a class of interacting particle systems on $\mathbb{Z}^d$ in which, as time evolves, particles execute random walks, produce offspring (on neighbouring sites) and (instantaneously) disappear when they meet other particles.…

In this work we address the problem of determining the asymptotic spectral measure of the product of independent, Gaussian random matrices with correlated entries, as the dimension and the number of multiplicative terms goes to infinity. More specifically, let $\{X_p(N)\}_{p=1}^\infty$…

Let $(e^{tA})_{t\geq0}$ be a $C_0$-contraction semigroup on a $2$-smooth Banach space $E$, let $(W_t)_{t\geq0}$ be a cylindrical Brownian motion in a Hilbert space $H$, and let $(g_t)_{t\geq0}$ be a progressively measurable process with values in the space $\gamma(H,E)$ of all…