  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.
