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

Existence of weak solutions of stochastic wave equations with nonlinearities of a critical growth driven by spatially homogeneous Wiener processes is established in local Sobolev spaces and local energy estimates for these solutions are proved. A new method to construct…

We establish various small deviation inequalities for the extremal (soft edge) eigenvalues in the beta-Hermite and beta-Laguerre ensembles. In both settings, upper bounds on the variance of the largest eigenvalue of the anticipated order follow immediately.

Sharp constants in a (one-sided) Burkholder-Davis-Gundy type estimate for stochastic integrals in a 2-smooth Banach space are found. As a consequence, exponential tail estimates for stochastic convolutions are obtained via Zygmund's extrapolation theorem.