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Gaussian measures on the affine group: Uniqueness of embedding and supports. (English) Zbl 1027.60002

Summary: It is shown that a Gaussian measure on the affine group can be embedded only into a uniquely determined Gaussian semigroup. The starting point of the proof is the fact that a Gaussian Lévy process (i.e., a Lévy process with vanishing Lévy measure) on the affine group satisfies a certain stochastic differential equation. Moreover, we will give a complete description of supports of Gaussian measures on the affine group using Siebert’s support formula.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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