Löwner's equation from a noncommutative probability perspective

Using concepts of noncommutative probability we show that the Löwner's evolution equation can be viewed as providing a map from paths of measures to paths of probability measures. We show that the fixed point of the Löwner map is the convolution semigroup of the semicircle law in the chordal case, and its multiplicative analogue in the radial case. We further show that the Löwner evolution "spreads out" the distribution and that it gives rise to a Markov process.