765 research outputs found
Asymptotics for Lipschitz percolation above tilted planes
We consider Lipschitz percolation in dimensions above planes tilted by
an angle along one or several coordinate axes. In particular, we are
interested in the asymptotics of the critical probability as as
well as Our principal results show that the convergence of
the critical probability to 1 is polynomial as and In addition, we identify the correct order of this polynomial
convergence and in we also obtain the correct prefactor.Comment: 23 pages, 1 figur
Correction to: Convergent numerical approximation of the stochastic total variation flow
We correct two errors in our paper [4]. First error concerns the definition
of the SVI solution, where a boundary term which arises due to the Dirichlet
boundary condition, was not included. The second error concerns the discrete
estimate [4, Lemma 4.4], which involves the discrete Laplace operator. We
provide an alternative proof of the estimate in spatial dimension by
using a mass lumped version of the discrete Laplacian. Hence, after a minor
modification of the fully discrete numerical scheme the convergence in
follows along the lines of the original proof. The convergence proof of the
time semi-discrete scheme, which relies on the continuous counterpart of the
estimate [4, Lemma 4.4], remains valid in higher spatial dimension. The
convergence of the fully discrete finite element scheme from [4] in any spatial
dimension is shown in [3] by using a different approach.Comment: Stoch PDE: Anal Comp (2022
Convergent numerical approximation of the stochastic total variation flow with linear multiplicative noise: the higher dimensional case
We consider fully discrete finite element approximation of the stochastic
total variation flow equation (STVF) with linear multiplicative noise which was
previously proposed in \cite{our_paper}. Due to lack of a discrete counterpart
of stronger a priori estimates in higher spatial dimensions the original
convergence analysis of the numerical scheme was limited to one spatial
dimension, cf. \cite{stvf_erratum}. In this paper we generalize the convergence
proof to higher dimensions
Convergent numerical approximation of the stochastic total variation flow
We study the stochastic total variation flow (STVF) equation with linear
multiplicative noise. By considering a limit of a sequence of regularized
stochastic gradient flows with respect to a regularization parameter
we obtain the existence of a unique variational solution of the
STVF equation which satisfies a stochastic variational inequality. We propose
an energy preserving fully discrete finite element approximation for the
regularized gradient flow equation and show that the numerical solution
converges to the solution of the unregularized STVF equation. We perform
numerical experiments to demonstrate the practicability of the proposed
numerical approximation
On the random dynamics of Volterra quadratic operators
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.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 of the system equals the set of all vertices. The convergence proof relies on a martingale-type limit theorem, which we prove in the appendix.DFG, GSC 14, Berlin Mathematical Schoo
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