Harnack's inequality for stable Lévy processes
It is proved that for a Lévy process \(\{X_{t},t>0\}\) on \(\mathbb R^{d}\) with density function \(f_{\nu}\) of the Lévy measure \(\nu\) satisfying the condition \(f_{\nu}(y)\leq M| y| ^{-d-\alpha},\;y\in \mathbb R^{d},\) there exists a constant \(C_1(\alpha,\nu)\) such that, for every nonegative function \(u\), harmonic in the ball \(B(0,1)\), the following Harnack inequality holds: \[ u(x_1)\leq C_1 u(x_2),\quad x_1,x_2\in B(0,1/2), \] if and only if there is a constant \(C_2\) such that \[ \int_{B(y,1/2)} | y-\nu|^{\alpha-d} f_\nu(\nu)d\nu\leq C_2\int_{B(y,1/2)}f_\nu(\nu)d\nu,\quad | y|\geq1. \] No boundedness from below for the density \(f_\nu\) is required. The proof is based on the following estimate of the Green function \(G(x,y)\) of the unit ball \(B(0,1)\) for the process \(X_{t}\): there is a constant \(C\) such that \[ C^{-1}S(\nu)|\nu-x|^{\alpha-d}\leq G(x,y)\leq CS(\nu)| \nu-x|^{\alpha-d}\quad \text{for }| x| <1/2\text{ and }\nu\in B(0,1). \] Here \(S(\nu)=E^{\nu}\tau_{B}\), and \(\tau_{B}\) is the first exit time from \(B\).
scientific article; zbMATH DE number 2162535
scientific article