We investigate a possible definition of expectation and conditional expectation for random variables with values in a local field such as the $p$-adic numbers. We define the expectation by analogy with the observation that for real-valued random variables in $L^2$…

We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the…

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time $n$ is $I+G/n^z$ where $G$ is a ``generator'' matrix, that is $G(i,j)>0$ for $i,j$ distinct, and $G(i,i)= -\sum_{k\ne i} G(i,k)$, and $z>0$ is a strength parameter. In these chains,…

We consider an irreducible pure jump Markov process with rates $Q=(q(x,y))$ on $\Lambda\cup\{0\}$ with $\Lambda$ countable and $0$ an absorbing state. A {\em quasi stationary distribution \rm} (QSD) is a probability measure $\nu$ on $\Lambda$ that satisfies: starting with $\nu$,…

Rates of molecular evolution along phylogenetic trees are influenced by mutation, selection and genetic drift. Provided that the branches of the tree correspond to lineages belonging to genetically isolated populations (e.g., multi-species phylogenies), the interplay between these three processes can…

In this paper we study typical distances in random graphs with i.i.d. degrees of which the tail of the common distribution function is regularly varying with exponent $1-\tau$. Depending on the value of the parameter $\tau$ we can distinct three…

An acyclic mapping from an $n$ element set into itself is a mapping $\varphi$ such that if $\varphi^k(x) = x$ for some $k$ and $x$, then $\varphi(x) = x$. Equivalently, $\varphi^\ell = \varphi^{\ell+1} = \ldots$ for $\ell$ sufficiently large. We…

Let $X_1, X_2, \dots$ be independent and symmetric random variables such that $S_n = X_1 + \cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = \sup_{1 \leq n \leq \infty} S_n$ (which is…