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In particular, if a random attractor is characterized by a family of invariant random compact sets, we show the equivalence among the asymptotic stability of this family, the Morse decomposition of the random attractor, and the existence of a random Lyapunov function.<\/jats:p>","DOI":"10.1137\/120862752","type":"journal-article","created":{"date-parts":[[2012,12,18]],"date-time":"2012-12-18T12:39:40Z","timestamp":1355834380000},"page":"1817-1847","source":"Crossref","is-referenced-by-count":10,"title":["Gradient Infinite-Dimensional Random Dynamical Systems"],"prefix":"10.1137","volume":"11","author":[{"given":"Tom\u00e1s","family":"Caraballo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jos\u00e9 A.","family":"Langa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenxin","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2012,12,18]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/24\/7\/010"},{"key":"atypb2","doi-asserted-by":"crossref","unstructured":"L. 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