PUB - Publikationen an der Universität Bielefeld

This thesis is motivated by stochastic particle systems arising in self-organized criticality, which are also known as "sandpile models'". As observed by Bak, Tang and Wiesenfeld (Phys. Rev. Lett. 59 (1987)), these systems stand out due to the fact that they converge without specific external tuning to a state in which power-law distributed intermittent events occur.

The present thesis aims to contribute to the mathematical understanding of this behaviour and of the underlying models in general by making them accessible to analytical tools. To this end, it is rigorously shown that under a suitable rescaling, modified sandpile models on finer and finer one-dimensional grids converge to the solutions of a stochastic partial differential equation (SPDE) with a singular-degenerate drift, driven by space-time white noise. Furthermore, the well-posedness of more general SPDEs of similar type is proved. Finally, the long time behaviour of solutions to the continuum limit SPDE is addressed by proving that the corresponding Markov process possesses a unique invariant measure.