Uniform Conditional Ergodicity and Intrinsic Ultracontractivity

We study two properties of semigroups of sub-Markov kernels, namely uniform conditional ergodicity and intrinsic ultracontractivity. In this paper we investigate the relationship between these two properties and we provide sufficient criteria as well as characterisations of them. In particular, our considerations show that, under suitable assumptions, the second property implies the first one. We also introduce a property called compact domination and show how this property and the parabolic boundary Harnack principle are related to the aforementioned properties. Furthermore, we apply these results in some special cases.