Functional inequalities for time-changed symmetric \(\alpha \)-stable processes
For \(\alpha\in (0, 2),\) let \(Z := (Z_t)_{t>0}\) be a (rotationally invariant) symmetric \(\alpha\)-stable process on \(\mathbb R^d,\) whose infinitesimal generator is given by the fractional Laplacian \(-(-\Delta)^{\alpha/2}.\) The authors consider time-changed symmetric \(\alpha\)-stable processes and establish sharp functional inequalities for such processes with \(d > 1\), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function \(W(x) = (1+|x|)^\beta\) with \(\beta > \alpha\), optimal Nash-type inequalities are obtained, which in turn give optimal upper bounds for the density function of the associated semigroups.
scientific article; zbMATH DE number 7379093
scientific article