By M G Kuhn
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The conjugacy of stochastic and random differential equations and the existence of global attractors. Preprint (1998) 24. : Attractors for hyperbolic stochastic partial differential equations. PhD thesis, in preparation 25. : Numerical approximation of random attractors. In Stochastic Dynamics, Springer, 13erlin Heidelberg New York (1999) 93 115 26. rivial noise. S. l'vI. ica 1(26) (1998) 13-51 27. o random equations. eme, Bremen (1999) 28. Kloeden, P. ion: the autonomous case. DANSE preprint 22/99 (1999) 29.
Fm which are contractive, that is ri < 1, a set A C;;; X is called se(f-slrnilaT with respect to the fi if it solves the equation A = iI(A) u ... U fm(A) . The idea is that A consists of pieces which are geometrically similar to A. Substituting A on the right-hand side again by the union of pieces, we see that A consists of even smaller pieces f;fj (A) etc. It is also rather easy to verify that among the compact subsets of X there is exactly one solution of the above equation. Nevertheless, if one takes arbitrary similarity maps, the resulting A is not very spectacular.
However, if (J" increases. 21). This is due to a transversal intersection of the stable and the unstable Gunter Ochs 22 manifold of the "perturbed origin" (the hyperbolic fixed point persists if the perturbation does not exceed a certain size). e. we have the equation dx dy with f f(x) = = = ydt (f(x) -rY) dt + c(x) 0 dW = -U' and c : IR --+ IR. In particular we will focus on the case ax - f3x 3 and c(x) == (J. The corresponding Markov process has a unique invariant measure p with density c exp ( ~1 E(x, y)) = c exp ( ~1 (U (x) + Y22) ) with respf;ct to Lebesgue measure, where c > 0 is a normalization constant 2 am1 T} -- u2')" Note that in particular this measure is independent of r if we choose (J = ,j2T}r with fixed T} > O.