Potential Theory for Transient Convolution Semigroups

Abstract

Let (µ_t)_t>0 be a convolution semigroup on the LCA-group G, and let (P _t )_t>0 denote the induced Feller semigroup. It is often useful, in the study of (µ_t)_t>0, to replace the potential operator $N = \smallint _o^\infty {\text{ }}{{\text{P}}_s}ds{\text{ for (}}{{\text{P}}_t}{{\text{)}}_{t >0}}$ with the measure (if it exists)

$$\kappa = \int\limits_0^\infty {\mu _t dt.} $$