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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 Bartłomiej 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 $\lim _{|x| \to \infty } V(x) = \infty$, then for every $t >0$ the operator $T_t$ is compact. We consider the class $\mathcal {V}$ of potentials $V$ such that $V \ge 0$, $\lim _{|x| \to \infty } V(x) = \infty$ 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 $\lim _{|x| \to \infty } V(x)/|x| = \infty$. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction $\phi _1$ for $T_t$. In particular, when $V(x) = |x|^{\beta }$, $\beta > 0$, 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 \[ \exp (-m^{1/{\alpha }}|x|) (|x| + 1)^{(-d - \alpha - 2 \beta -1 )/2}.\]

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Additional Information

Tadeusz Kulczycki
Affiliation: Institute of Mathematics, Wrocław University of Technology, Wyb. Wyspianskiego 27, 50-370 Wrocław, Poland

Bartłomiej Siudeja
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906

Keywords: Intrinsic ultracontractivity, relativistic, Feynman-Kac semigroup, Schrödinger 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.