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, M. ; 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...