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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes
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by Tadeusz Kulczycki and Bartłomiej Siudeja PDF
Trans. Amer. Math. Soc. 358 (2006), 5025-5057 Request permission

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
  • Email: tkulczyc@im.pwr.wroc.pl
  • Bartłomiej Siudeja
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
  • Email: siudeja@math.purdue.edu
  • 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
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 5025-5057
  • MSC (2000): Primary 47G30, 60G51
  • DOI: https://doi.org/10.1090/S0002-9947-06-03931-6
  • MathSciNet review: 2231884