publication . Article . 2006

Fractal dimension of a random invariant set

José A. Langa; James C. Robinson;
Open Access
  • Published: 01 Feb 2006 Journal: Journal de Mathématiques Pures et Appliquées, volume 85, issue 2, pages 269-294 (issn: 0021-7824, Copyright policy)
  • Publisher: Elsevier BV
Abstract
Abstract In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of t...
Subjects
free text keywords: Applied Mathematics, Mathematics(all), Random invariant set, Fractal dimension, Parabolic equations, General Mathematics, Correlation dimension, Fractal, Effective dimension, Hausdorff dimension, Minkowski–Bouligand dimension, Mathematical analysis, Poincaré recurrence theorem, Finite set, Mathematics
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