Monotone increment processes, classical Markov processes, and Loewner chains
Volume 552 / 2020
Dissertationes Mathematicae 552 (2020), 1-119
MSC: Primary 30C35, 46L53, 60G51; Secondary 30C55, 30C80, 46L54, 46N30, 60E07, 60E10, 81R15, 81S25.
DOI: 10.4064/dm808-1-2020
Published online: 31 August 2020
Abstract
We prove one-to-one correspondences between certain decreasing Loewner chains in the upper half-plane, a special class of real-valued Markov processes, and quantum stochastic processes with monotonically independent additive increments. This leads us to a detailed investigation of probability measures on $\mathbb R$ with univalent Cauchy transform. We discuss several subclasses of such measures and obtain characterizations in terms of analytic and geometric properties of the corresponding Cauchy transforms.
Furthermore, we obtain analogous results for the setting of decreasing Loewner chains in the unit disk, which correspond to quantum stochastic processes of unitary operators with monotonically independent multiplicative increments.
