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Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes
Authors:Tadeusz Kulczycki   Bartlomiej Siudeja
Affiliation: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 $ 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 xvert 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 xvert 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 xvert to infty} V(x)/vert xvert = 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 xvert^{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

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

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