A Harnack-type inequality for non-symmetric Markov semigroups
Consider on a locally compact Polish space \(E\) a strong Feller semigroup \(T_t\) which is irreducible, i.e., \(T_t1_G >0\) for any \(t>0\), \(x\in E\) and any open set \(\emptyset \neq G\subset E\). The author shows that \(T_t\) satisfies a Harnack-type inequality of the form \(\sup_E T_tu \leq c_t \inf_E T_tu\) for \(t>0\) and positive measurable functions \(u\) if and only if \(T_t\) admits a unique invariant probability measure \(\pi\) and if \(T_t\) is ultracontractive with respect to the \(L^p\)-scale defined by \(\pi\). For diffusion semigroups on Riemannian manifolds with Ricci curvature bounded from below, the constant \(c_t\) appearing in the Harnack inequality is explicitly calculated and a new condition for ultracontracitvity (hence the Harnack inequality) to hold is given.
scientific article; zbMATH DE number 5072841