Concentration of the first eigenfunction for a second order elliptic operator |
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Authors: | David Holcman Ivan Kupka |
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Institution: | 1. Department of Mathematics, Weizmann Institute of Science 76100 Rehovot, Israel;2. Département de mathématiques, 175, rue du Chevaleret, 75013 Paris, France |
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Abstract: | We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold when the diffusion constant ε goes to zero. We assume that the first order term is given by a vector field b, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure Y. Kifer, Principal eigenvalues, topological pressure and stochastic stability of equilibrium states, Israel J. Math. 70 (1990) (1) 1–47] is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis. To cite this article: D. Holcman, I. Kupka, C. R. Acad. Sci. Paris, Ser. I 341 (2005). |
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