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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes


Authors: Tadeusz Kulczycki and Bartlomiej Siudeja
Journal: Trans. Amer. Math. Soc. 358 (2006), 5025-5057
MSC (2000): Primary 47G30, 60G51
Published electronically: June 13, 2006
MathSciNet review: 2231884
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Abstract: Let $ X_t$ be the relativistic $ \alpha$-stable process in $ \mathbf{R}^d$, $ \alpha \in (0,2)$, $ d > \alpha$, with infinitesimal generator $ H_0^{(\alpha)}= - ((-\Delta +m^{2/\alpha})^{\alpha/2}-m)$. We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup $ T_t$ for this process with generator $ H_0^{(\alpha)} - V$, $ V \ge 0$, $ V$ locally bounded. We prove that if $ \lim_{\vert x\vert \to \infty} V(x) = \infty$, then for every $ t >0$ the operator $ T_t$ is compact. We consider the class $ \mathcal{V}$ of potentials $ V$ such that $ V \ge 0$, $ \lim_{\vert x\vert \to \infty} V(x) = \infty$ and $ V$ is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For $ V$ in the class $ \mathcal{V}$ we show that the semigroup $ T_t$ is IU if and only if $ \lim_{\vert x\vert \to \infty} V(x)/\vert x\vert = \infty$. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction $ \phi_1$ for $ T_t$. In particular, when $ V(x) = \vert x\vert^{\beta}$, $ \beta > 0$, then the semigroup $ T_t$ is IU if and only if $ \beta >1$. For $ \beta >1$ the first eigenfunction $ \phi_1(x)$ is comparable to

$\displaystyle \exp(-m^{1/{\alpha}}\vert x\vert) \, (\vert x\vert + 1)^{(-d - \alpha - 2 \beta -1 )/2}.$


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Additional Information

Tadeusz Kulczycki
Affiliation: Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspianskiego 27, 50-370 Wrocław, Poland
Email: tkulczyc@im.pwr.wroc.pl

Bartlomiej Siudeja
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
Email: siudeja@math.purdue.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-06-03931-6
PII: S 0002-9947(06)03931-6
Keywords: Intrinsic ultracontractivity, relativistic, Feynman-Kac semigroup, Schr\"odinger operator, first eigenfunction
Received by editor(s): March 23, 2004
Received by editor(s) in revised form: November 11, 2004
Published electronically: June 13, 2006
Additional Notes: This work was supported by KBN grant 2 P03A 041 22 and RTN Harmonic Analysis and Related Problems, contract HPRN-CT-2001-00273-HARP
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.