Analytic Semigroups and Degenerate Elliptic Operators with Unbounded Coefficients: A Probabilistic Approach
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چکیده
In the present paper we are dealing with generation of analytic semigroups in the space of continuous and bounded functions by suitable second order differential operators, which have unbounded coefficients and may be degenerate. This means that we are away from the classical framework in the study of the generation of analytic semigroups by elliptic operators (see Lunardi [25] for a comprehensive overview on the more recent results in this field) under two respects: firstly, we are able to overcome the usual assumption of boundedness of coefficients; secondly our results adapt to a wide class of degenerate operators. Moreover, we do not use the more classical deterministic techniques developed beginning from the works of Stewart [27] and [28]. Actually, we regard our operators as the diffusion operators corresponding to suitable stochastic differential equations and hence we proceed by giving an explicit probabilistic representation of the semigroups. The study of partial differential equations by probabilistic methods is classical by now. Starting with the books by Strook and Varadhan [29], Ethier and Kurtz [16] and others, many results have been proved about existence, uniqueness and regularity. These results have been extended in various aspects, including less growth restrictions and less regularity for the coefficients, as well as more degeneracy. To this purpose it is worthwhile to cite the interesting books by Friedlin [20] and Krylov [23]. All these doi:10.1006 jdeq.2000.3788, available online at http: www.idealibrary.com on
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تاریخ انتشار 2000