We give a criterion for the non-reconstructability of tree-indexed $q$-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix $M$. Non-reconstruction holds if $c(M)$ times the expected number of offspring on the Galton-Watson tree…

We consider particle systems in $R$ with initial configurations belonging to a class of measures that obey a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of…

Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$,…

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…

Let $X_{1},X_{2},...$ denote independent, identically distributed random variables with common distribution $F$, and $S$ the corresponding random walk with $\rho :=\lim_{n\rightarrow \infty }P(S_{n}>0)$ and $\tau :=\inf \{n\geq 1:S_{n}\leq 0\}$. We assume that $X$ is in the domain of attraction of…

Dyson's Brownian motion model with the parameter $\beta=2$, which we simply call the Dyson model in the present paper, is realized as an $h$-transform of the absorbing Brownian motion in a Weyl chamber of type A. Depending on initial configuration with a finite number of…