Conference Agenda

In this joint work we consider a general one-dimensional particle system. These processes are mainly characterized by multiplicity parameters controlling the strength of the particles' interaction. It is well-known that collisions between particles never take place when all of these multiplicities are large, but occur almost surely otherwise. It was recently shown by the authors that the collision times in special cases the multivariate Bessel processes of rational type is a piecewise-linear function of the minimum of its multiplicities, but is independent of the dimension of the process’s domain. This implies that the Hausdorff dimension does not depend on the particle number. In this talk, we present an approach to extend this result to a general particle system.

For some discretely observed path of oscillating Brownian motion with level of self-organized criticality $\rho_0$, we prove in the infill asymptotics that the MLE is $n$-consistent, where $n$ denotes the sample size, and derive its limit distribution with respect to stable convergence. As the transition density of this homogeneous Markov process is not even continuous in $\rho_0$, interesting and somewhat unexpected phenomena occur: The likelihood function splits into several components, each of them contributing very differently depending on how close the argument $\rho$ is to $\rho_0$. Correspondingly, the MLE is successively excluded to lay outside a compact set, a $1/\sqrt{n}$-neighborhood and finally a $1/n$-neigborhood of $\rho_0$ asymptotically. Sequentially and as a process in $\rho$, the martingale part of the suitably rescaled local log-likelihood function exhibits a bivariate Poissonian behavior in the stable limit with its intensity being a function of the local time at $\rho_0$.

We condition a Brownian motion on having an atypically small $L_2$-norm on a long time interval. The obtained limiting process is an Ornstein-Uhlenbeck process. As a main ingredient, we prove a result on the small ball probabilities of non-centered Brownian motion.

Most statistical methods for stochastic partial differential equations (SPDEs) based on discrete observations are limited to one space dimension or to quite restrictive settings. In order to study SPDEs on a bounded domain driven by a stochastic noise process which is white in time and possibly colored in space, we aim for bridging the gap between two popular observations schemes studied for statistics for SPDEs, namely, discrete observations and local measurements. To this end, we have to extend the local measurements approach to kernels of distribution type. This link allows us to construct a non-parametric estimator for the diffusivity based on discrete high-frequency observations. The talk is based on joint work with Randolf Altmeyer and Florian Hildebrandt.