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On Adjoint Additive Processes

Evans Kristian P., Jacob Niels

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 2

205--223

Starting with an additive process (Yt)t≥0, it is in certain cases possible to construct an adjoint process (Xt)t≥0 which is itself additive. Moreover, assuming that the transition densities of (Yt)t≥0 are controlled by a natural pair of metrics dψ;t and δψ;t, we can prove that the transition densities of (Xt)t≥0 are controlled by the metrics δ ψ,1/treplacing dψ,t and dψ,/treplacing δψ,t.

Musielak−Orlicz−Sobolev spaces on arbitrary metrique space

Aissaoui N., Akdim Y., Hassib M. C.

Commentationes Mathematicae

2016

Vol 56, No. 2

169--183

In this article we define Musielak−Orlicz−Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each Musielak−Orlicz−Sobolev function has a quasi-continuous representative.