Free Brownian motion and free convolution semigroups: multiplicative case

Abstract

Abstract. We consider μ, ν a pair of probability measures on the unit circle such that Σλ(ην(z)) = z/ημ(z). We prove that the same type of equation holds for any t ≥ 0 when we replace ν by ν λt and μ by Mt(μ), where λt is the analogue of the normal distribution with respect to and Mt is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of measures associated with subordination functions in free Brownian motion and free convolution semigroups. We use the modified S-transform introduced by Raj Rao and Speicher to deal with the case that ν has zero mean. The same type of the result holds for convolutions on the positive real line. We also obtain some regularity properties for the free multiplicative analogue of the normal distributions.