Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes |
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Authors: | Tadeusz Kulczycki Bartlomiej Siudeja |
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Affiliation: | Institute of Mathematics, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland ; Department of Mathematics, Purdue University, West Lafayette, Indiana 47906 |
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Abstract: | Let be the relativistic -stable process in , , , with infinitesimal generator . We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup for this process with generator , , locally bounded. We prove that if , then for every the operator is compact. We consider the class of potentials such that , and is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For in the class we show that the semigroup is IU if and only if . If this condition is satisfied we also obtain sharp estimates of the first eigenfunction for . In particular, when , , then the semigroup is IU if and only if . For the first eigenfunction is comparable to |
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Keywords: | Intrinsic ultracontractivity, relativistic, Feynman-Kac semigroup, Schr" odinger operator, first eigenfunction |
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