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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Contraction semigroups for diffusion with drift


Author: R. Seeley
Journal: Trans. Amer. Math. Soc. 283 (1984), 717-728
MSC: Primary 58G32; Secondary 35K15
DOI: https://doi.org/10.1090/S0002-9947-1984-0737895-3
MathSciNet review: 737895
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Abstract: Recently Dodziuk, Karp and Li, and Strichartz have given results on existence and uniqueness of contraction semigroups generated by the Laplacian $ \Delta $ on a manifold $ M$; earlier, Yau gave related results for $ L = \Delta + V$ for a vector field $ V$. The present paper considers $ L = \Delta - V - c$, with $ c$ a real function, and gives conditions for (a) uniqueness of semigroups on the bounded continuous functions, (b) preservation of $ {C_0}$ (functions vanishing at $ \infty $) by the minimal semigroup, and (c) existence and uniqueness of contraction semigroups on $ {L^p}(\mu ),\;1 \leqslant p < \infty $, for an arbitrary smooth density $ \mu $ on $ M$. The conditions concern $ L\rho /\rho $, where $ \rho $ is a smooth function, $ \rho \to \infty $ as $ x \to \infty $. They variously extend, strengthen, and complement the previous results mentioned above.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0737895-3
Article copyright: © Copyright 1984 American Mathematical Society