Abstract
Let ϕ be a submultiplicative function on a locally compact group G and let S be a convolution semigroup on G with Lévy measure η. It is shown that the measures of S integrate ϕ if and only ifη integrates ϕ outside some neighbourhood of the identity of G (Theorem 1). Moreover if (X(t);t≥0) is the G-valued process with independent increments associated with the semigroup S it is shown that the measures of S integrate ϕ if and only if the random variable sup {ϕ (X(t)):0≤t≤1} is integrable (Theorem 2).
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Siebert, E. Continuous convolution semigroups integrating a submultiplicative function. Manuscripta Math 37, 383–391 (1982). https://doi.org/10.1007/BF01166228
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DOI: https://doi.org/10.1007/BF01166228