Monotone increment processes, classical Markov processes, and Loewner chains

2D growth processes: SLE and Loewner chains

This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions (SLE) which are Markov processes describing interfaces in 2D critical systems. It starts with an informal discussion, using numerical simulations, of variou...

Markov Chains and Jump Processes

Markov Property of Monotone Lévy Processes

Monotone Lévy processes with additive increments are defined and studied. It is shown that these processes have natural Markov structure and their Markov transition semigroups are characterized using the monotone Lévy-Khintchine formula. 17 Monotone Lévy processes turn out to be related to classical Lévy processes via Attal's " remarkable transformation. " A monotone analogue of the family of e...

Markov Chains and De-initialising Processes

We define a notion of de-initialising Markov chains. We prove that to analyse convergence of Markov chains to stationarity, it suffices to analyse convergence of a deinitialising chain. Applications are given to Markov chain Monte Carlo algorithms and to convergence diagnostics.

Recombination Processes and Nonlinear Markov Chains

Bacteria are known to exchange genetic information by horizontal gene transfer. Since the frequency of homologous recombination depends on the similarity between the recombining segments, several studies examined whether this could lead to the emergence of subspecies. Most of them simulated fixed-size Wright-Fisher populations, in which the genetic drift should be taken into account. Here, we u...