Let $\{X_t^\mu,t\geq0\}$ be a Lévy process on $\mathbb{R}^d$ whose distribution at time $1$ is a $d$-dimensional infinitely distribution $\mu$. It is known that the set of all infinitely divisible distributions on $\mathbb{R}^d$, each of which is represented by the law…

In this paper, we consider a quasi-linear stochastic heat equation with spatial dimension one, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a…

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension $d\geq2$. These include a scale invariant version of the classical (Poisson) Boolean…

We consider a model of a discrete time "interacting particle system" on the integer line where infinitely many changes are allowed at each instance of time. We describe the model using chameleons of two different colours, viz., red (R) and…

Consider the center of mass of a supercritical branching-Brownian motion. In this article we first show that it is a Brownian motion being slowed down such that it tends to a limiting position almost surely, and that this is also…

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein's method, as well as the use of appropriate discrete Malliavin operators. As the bounds are given in terms of Malliavin operators, no…

The Adaptive Metropolis (AM) algorithm is based on the symmetric random-walk Metropolis algorithm. The proposal distribution has the following time-dependent covariance matrix, at step $n+1$, $S_n=\mathrm{Cov}(X_1,\ldots,X_n)+\varepsilon I$, that is, the sample covariance matrix of the history of the chain plus…

One can construct a sequence of rescaled perturbations of voter processes in dimension $d=1$ whose approximate densities are tight. By combining both long-range models and fixed kernel models in the perturbations and considering the critical long-range case, results of Cox…

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every…

In a companion paper (M. Birkner, A. Greven, F. den Hollander, Quenched LDP for words in a letter sequence, Probab. Theory Relat. Fields 148, no. 3/4 (2010), 403-456), a quenched large deviation principle (LDP) has been established for the empirical…