Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals

Markus Böhm; Björn Schmalfuss

doi:10.3934/dcdsb.2018303

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2019, Volume 24, Issue 7: 3115-3138. Doi: 10.3934/dcdsb.2018303

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Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals

Markus Böhm and
Björn Schmalfuss^,

Friedrich Schiller University, Institute of Mathematics, Ernst-Abbe-Platz 2, 07743, Jena, Germany

* Corresponding author: Björn Schmalfuss
* Corresponding author: Björn Schmalfuss

Dedicated to Peter E. Kloeden on Occasion of his Seventieth Birthday

Received: April 2018

Early access: October 2018

Published: May 2019

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Abstract

We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.

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Mathematics Subject Classification: Primary: 37H15, 37L55; Secondary: 60H15, 35R60, 60J65, 34D45.

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Figure 1. An approximation of the Sierpinski gasket using a sequence of graphs

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