Abstract
When a system of ordinary differential equations on ℜd have an attractor [Formula: see text] given as the omega-limit set of a compact set [Formula: see text], then provided that D contains [Formula: see text] necessarily [Formula: see text], i.e. the boundary of the attractor is the omega-limit set of the boundary of D. In general only a weaker result, [Formula: see text], can be shown to hold for random attractors arising from systems of stochastic ordinary differential equations, where now D is any compact set such that [Formula: see text] is contained in D with a positive probability. However, if in addition [Formula: see text] is empty with positive probability, then in fact [Formula: see text].
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