Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Tadeusz Kulczycki Bartlomiej Siudeja

Institution:	Institute of Mathematics, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wroclaw, Poland ; Department of Mathematics, Purdue University, West Lafayette, Indiana 47906

Abstract:	Let 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 for this process with generator $H_0^{(\alpha)} - V$ , $V \ge 0$ , locally bounded. We prove that if $\lim_{\vert x\vert \to \infty} V(x) = \infty$ , then for every the operator is compact. We consider the class $\mathcal{V}$ of potentials such that $V \ge 0$ , $\lim_{\vert x\vert \to \infty} V(x) = \infty$ and is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For in the class $\mathcal{V}$ we show that the semigroup 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 . In particular, when $V(x) = \vert x\vert^{\beta}$ , $\beta > 0$ , then the semigroup is IU if and only if $\beta >1$ . For $\beta >1$ the first eigenfunction $\phi_1(x)$ is comparable to $\displaystyle \exp(-m^{1/{\alpha}}\vert x\vert) \, (\vert x\vert + 1)^{(-d - \alpha - 2 \beta -1 )/2}.$

Keywords:	Intrinsic ultracontractivity relativistic Feynman-Kac semigroup Schr\"odinger operator first eigenfunction

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