We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation…

Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d \le a)$ due to Gikhman and Kiefer for $d = 1,2, \ldots$ is shown to be valid for all real $d >0$.…

For a quasi-regular (symmetric) Dirichlet space $( {\cal E}, {\cal F})$ and an associated symmetric standard process $(X_t, P_x)$, we show that, for $u in {\cal F}$, the additive functional $u^*(X_t) - u^*(X_0)$ is a semimartingale if and only if…

We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the…

We construct and study space homogeneous and isotropic random measures (MMRM) which generalize the so-called MRM measures constructed by previous authors. Our measures satisfy an exact scale invariance equation and are therefore natural models in dimension 3 for the dissipation…

We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). We assume that the distribution of the entries have a Gaussian component with variance $N^{-3/4+\beta}$ for some positive $\beta>0$. We prove that the local eigenvalue statistics…

We study a one dimensional two-type contact process with equal rate of propagation (and death) of the two types. We show that the progeny of a finite number of mutants has a positive probability of survival if and only at…

We consider the internal diffusion limited aggregation (IDLA) process on the infinite cluster in supercritical Bernoulli bond percolation on $\mathbb{Z}^d$. It is shown that the process on the cluster behaves like it does on the Euclidean lattice, in that the…

We investigate a functional limit theorem (homogenization) for Reflected Stochastic Differential Equations on a half-plane with stationary coefficients when it is necessary to analyze both the effective Brownian motion and the effective local time. We prove that the limiting process…

We construct martingales whose 1-dimensional marginals are those of a centered self-decomposable variable multiplied by some power of time $t$. Many examples involving quadratic functionals of Bessel processes are discussed