Loewner's equation in noncommutative probability

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Using concepts of noncommutative probability we show that the Loewner'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 Loewner 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 Loewner evolution ``spreads out'' the distribution and that it gives rise to a Markov process.