# Anomalous Stochastic Processes

If the time-evolution of the mean-squared displacement of some quantity is non-linear,

the system is said to exhibit anomalous diffusion. The underlying mechanisms leading to such anomalous diffusion can be multifold. Our group focuses on processes whose

anomalous behavior is due to heavy-tailed distributions of either the waiting time

distribution between the displacements or of the displacements themselves.

One topic of our group is the path integral description of Continuous Time Random Walks (CTRWs) which are random walks where the time between successive displacements is random and governed by some probability distribution. While a path integral formulation of CTRWs shows not only an interesting mathematical structure, it furthermore enables the application of many sophisticated approximation schemes for CTRWs in non-linear potentials or with multiplicative noise, which were originally invented in the context of quantum field theory.

Furthermore we are interested in the properties of two-dimensional anomalous diffusion processes. Here we use the concept of the convex hull which is the minimal convex

polygon enclosing a sample trajectory. We apply our results to model the home-range

of certain foraging animals.

Non-exponential waiting times cannot only significantly alter the transport properties

of complex systems but also change the birth and death dynamics of constituents within the system itself. These considerations lead to anomalous reaction kinetics. We focus on the mathematical description of reactions which have one reaction channel being governed by anomalous, non-Poissonian kinetics while all other channels exhibit regular, Poissonian kinetics. The developed models are relevant in many biological systems such as gene transcription and ion channel dynamics.

**Contact:**Stephan Eule