arXiv subjects

Moritz Biskamp

Publications and source records attributed to Moritz Biskamp.

3 recordsLinked to original sources

arXiv2013

Asymptotic support theorem for planar isotropic Brownian flows

It has been shown by various authors that the diameter of a given nontrivial bounded connected set $\mathcal{X}$ grows linearly in time under the action of an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the precise deterministic linear growth rate K of the diameter is known to exist. In this paper we will extend this result to an asymptotic support theorem for the time-scaled trajectories of a planar IBF $φ$, which has a positive top-Lyapunov exponent, starting in a nontrivial compact connected set $\mathcal{X}\subseteq \mathbf{R}^2$; that is, we will show convergence in probability of the set of time-scaled trajectories in the Hausdorff distance to the set of Lipschitz continuous functions on [0,1] starting in 0 with Lipschitz constant K.

math.PR↗

arXiv2012

Pesin's Formula for Random Dynamical Systems on $R^d$

Pesin's formula relates the entropy of a dynamical system with its positive Lyapunov exponents. It is well known, that this formula holds true for random dynamical systems on a compact Riemannian manifold with invariant probability measure which is absolutely continuous with respect to the Lebesgue measure. We will show that this formula remains true for random dynamical systems on $R^d$ which have an invariant probability measure absolutely continuous to the Lebesgue measure on $R^d$. Finally we will show that a broad class of stochastic flows on $R^d$ of a Kunita type satisfies Pesin's formula.

math.PR↗

arXiv2012

Absolute Continuity Theorem for Random Dynamical Systems on $R^d$

In this article we provide a proof of the so called absolute continuity theorem for random dynamical systems on $R^d$ which have an invariant probability measure. First we present the construction of local stable manifolds in this case. Then the absolute continuity theorem basically states that for any two transversal manifolds to the family of local stable manifolds the induced Lebesgue measures on these transversal manifolds are absolutely continuous under the map that transports every point on the first manifold along the local stable manifold to the second manifold, the so-called Poincaré map or holonomy map. In contrast to known results, we have to deal with the non-compactness of the state space and the randomness of the random dynamical system.

math.PR↗