2022/03/01 by Huang, Hao-Wei, Wang, Jiun-Chau
#FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)
paper · doi:10.48550/arxiv.2203.00421
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 \boxplus 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 \boxplus 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 \boxplus 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.