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Continuous convolution hemigroups integrating a submultiplicative function

Hazod W.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

317--337

Unifying and generalizing previous investigations for vector spaces and for locally compact groups, E. Siebert obtained the following remarkable result: A Lévy process on a completely metrizable topological group G, resp. a continuous convolution semigroup (μt)t≥0 of probabilities, satisfies a moment condition ∫fdμt < ∞ for some submultiplicativefunction f > 0 if and only if the jump measure of the process, resp. the Lévy measure η of the continuous convolution semigroup, satisfies ∫CUfdη < ∞ for some neighbourhood U of the unit e. Here we generalize this result to additive processes, resp. convolution hemigroups (μs;t)s≤t, on (second countable) locally compact groups.