**Connectedness of random set attractors**

*Scheutzow, Michael ; Vorkastner, Isabell* (2018-12-28)

We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup, however, connectedness has only been shown under stronger connectedness assumptions on the state space. Under a weak continuity condition on the random dynam...

**A coupling approach to Doob’s theorem**

*Kulik, Alexei ; Scheutzow, Michael* (2015)

We provide a coupling proof of Doob’s theorem which says that the transition probabilities of a regular Markov process which has an invariant probability measure μ converge to μ in the total variation distance. In addition we show that non-singularity (rather than equivalence) of the transition probabilities suffices to ensure convergence of the transition probabilities for μ-almost all initial...

**Pinning of interfaces in random media**

*Dirr, Nicolas ; Dondl, Patrick W. ; Scheutzow, Michael* (2011)

For a model for the propagation of a curvature sensitive interface in a time independent random medium, as well as for a linearized version which is commonly referred to as Quenched Edwards– Wilkinson equation, we prove existence of a stationary positive supersolution at non-vanishing applied load. This leads to the emergence of a hysteresis that does not vanish for slow loading, even though th...

**Noise induced synchronization and related topics**

*Vorkastner, Isabell* (2018)

This thesis focuses on noise induced synchronization. Noise induced synchronization describes the stabilizing effect of noise on the long-time dynamics of a random dynamical system. While the attractor in the absence of noise is not a single point, the random attractor collapses to a single random point under the addition of noise. In the first part, we consider a system that is known to synch...

**Existence and uniqueness of solutions of stochastic functional differential equations**

*Renesse, Max-K. von ; Scheutzow, Michael* (2010)

Using a variant of the Euler–Maruyama scheme for stochastic functional differential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or “monotonicity”) conditions are sufficient for local existence and uniqueness of strong solutions. In case of explosion the method yields the maximal solution up to the explosion time. We also provide a we...

**Pinning of interfaces in a random elastic medium and logarithmic lattice embeddings in percolation**

*Dondl, Patrick W. ; Scheutzow, Michael ; Throm, Sebastian* (2015)

For a model of a driven interface in an elastic medium with random obstacles we prove the existence of a stationary positive supersolution at non-vanishing driving force. This shows the emergence of a rate-independent hysteresis through the interaction of the interface with the obstacles despite a linear (force = velocity) microscopic kinetic relation. We also prove a percolation result, namely...

**On the random dynamics of Volterra quadratic operators**

*Jamilov, U. U. ; Scheutzow, Michael ; Wilke-Berenguer, M.* (2015)

We consider random dynamical systems generated by a special class of Volterra quadratic stochastic operators on the simplex Sm-1. We prove that in contrast to the deterministic set-up the trajectories of the random dynamical system almost surely converge to one of the vertices of the simplex Sm-1, implying the survival of only one species. We also show that the minimal random point attractor o...