The convolution equation $\sigma*\mu=\mu$ on non-compact non-abelian semigroups
In probability theory, often in connection with problems on weak convergence, and also in other contexts, convolution equations of the form $\sigma*\mu=\mu$ come up. Many years ago, Choqet and Deny (C. R. Acad. Sci. Paris 250 (1960) 799-801) studied these equations in locally compact abelian groups. Later, Szekely and Zeng (J. Theoret. Probab. 3(2) (1990) 361-365) studied these equations in abelian semigroups. Like in , the results in  are also complete. Thus, these equations are studied here for the first time on non-compact non-abelian semigroups. Our main results are Theorems 3.1 and 3.3 in section 3. They are new results as far as we know, and also the best possible under a minor condition. All semigroups in this paper are, unless otherwise mentioned, locally compact Hausdorff second countable topological semigroups. Theorems 3.1 and 3.3 hold for these semigroups.
Volume 130, 2020
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