Abstract
We relate ergodicity, monotonicity and attractors of a random dynamical system (rds). Our first result states that an rds which is both monotone and ergodic has a weak random attractor which consists of a single point. Then we show that ergodicity alone is insufficient for the existence of a weak random attractor. In particular we present an rds in ℝd, d ≥ 2 namely an isotropic Brownian flow with drift, whose single-point motion is an ergodic diffusion process and which does not have a weak attractor. It seems that this is the first example of this kind in the literature.
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© 2007 Birkhäuser Verlag Basel/Switzerland
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Scheutzow, M. (2007). Attractors for Ergodic and Monotone Random Dynamical Systems. In: Dalang, R.C., Russo, F., Dozzi, M. (eds) Seminar on Stochastic Analysis, Random Fields and Applications V. Progress in Probability, vol 59. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-8458-6_18
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DOI: https://doi.org/10.1007/978-3-7643-8458-6_18
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