Monotone increment processes, classical Markov processes and Loewner chains (Mathematical aspects of quantum fields and related topics)

Abstract

This is a summary of the paper [FHS20]. The main result is the construction of bijections of the three objects: non-commutative stochastic processes with monotonically independent increments; certain decreasing Loewner chains in the upper half-plane; a special class of real-valued Markov processes. The Markov process associated with monotone Brownian motion coincides with the Azéma martingale when started at the position 0, and it has the arcsine distribution of mean O and variance t at time t.