We investigate small deviation properties of Gaussian random fields in the space $L_q(R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, i.e., those which are singular with respect to the…

The infinite divisibility of probability distributions on the space $P (R )$ of probability distributions on $R$ is defined and related fundamental results such as the Levy-Khintchin formula, representation of Ito type of infinitely divisible RPD, stable RPD and Levy…

Small world models are networks consisting of many local links and fewer long range `shortcuts', used to model networks with a high degree of local clustering but relatively small diameter. Here, we concern ourselves with the distribution of typical inter-point…

We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and $+\infty$. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of…

We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a dynamic risk measure time-consistent if it assigns to a process of…

We introduce a class of iterated processes called $\alpha$-time Brownian motion for $0<\alpha \leq 2$. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric $\alpha$-stable process. We prove a Chung-type law of the iterated…

We study a model of $ N $ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the…

We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $B =\{B(t)\}_{t \in R_+^N}$: (1) If $\text{dim}_H F \leq d/2$, then $B(F)$ is almost surely a Salem set. (2) If $N \leq d/2$,…

This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure $G$ of an open connected set in ${\mathbb…

Consider the following forest-fire model where trees are located on sites of $\mathbb{Z}$. A site can be vacant or be occupied by a tree. Each vacant site becomes occupied at rate $1$, independently of the other sites. Each site is…