On the finite dimensionality of random attractors¹

¹ This work was done while the author was visiting the Scuola Normale Superiore di Pisa, whose warm hospitality and generous support is gratefully acknowledged

Abstract

In this paper, we prove that in many cases the random attractor defined and in [2] and [4] has finite Hausdorff dimension. We generalize a method used in [8] and [1] in the determistic case, the generalization is made through an ergodicity argument. The main assumption amounts to say that the diameter of the attractor has finite expectation. We apply our results to a stochastic reaction-diffusion equation and to a stochastic nonlinear wave equation. Other examples could be considered

¹ This work was done while the author was visiting the Scuola Normale Superiore di Pisa, whose warm hospitality and generous support is gratefully acknowledged

¹ This work was done while the author was visiting the Scuola Normale Superiore di Pisa, whose warm hospitality and generous support is gratefully acknowledged

Notes

¹ This work was done while the author was visiting the Scuola Normale Superiore di Pisa, whose warm hospitality and generous support is gratefully acknowledged