Abstract
Given a free additive convolution semigroup (μt)t≥0 and a probability measure ν on R, we find the necessary and sufficient conditions for the process μt⊞ν to be Lebesgue absolutely continuous with a positive and analytic density throughout R at all time t>0. For semigroups without this property, we find the necessary and sufficient conditions for the density of μt⊞ν to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that μt⊞ν has a finite number of connected components in its support if both the Lévy measure of (μt)t≥0 and the initial law ν do.
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