Showing papers by "Jan M. Swart published in 2015"

The Brownian web, the Brownian net, and their universality

Abstract: The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briey survey models and results connected to the Brownian web and net, including alternative topologies, population genetic models, true self-repelling motion, planar aggregation, drainage networks, oriented percolation, black noise and critical percolation. Some open questions are discussed at the end.

A Course in Interacting Particle Systems

Abstract: These lecture notes give an introduction to the theory of interacting particle systems. The main subjects are the construction using generators and graphical representations, the mean field limit, stochastic order, duality, and the relation to oriented percolation. An attempt is made to give a large number of examples beyond the classical voter, contact and Ising processes and to illustrate these based on numerical simulations.

A particle system with cooperative branching and coalescence.

Abstract: nitely many particles a.s. ends up with a single particle. Both statements are not true for high branching rates. An interesting feature of the process is that the spectral gap is zero even for low branching rates. Indeed, if the branching rate is small enough, then we show that for the process started in the fully occupied state, the particle density decays as one over the square root of time, and the same is true for the decay of the probability that the process still has more than one particle at a later time if it started with two particles.

The limiting shape of a full mailbox

Abstract: We study a model for email communication due to Gabrielli and Caldarelli, where someone receives and answers emails at the times of independent Poisson processes with intensities $\lambda_{\rm in}>\lambda_{\rm out}$. The receiver assigns i.i.d. priorities to incoming emails according to some atomless law and always answers the email in the mailbox with the highest priority. Since the frequency of incoming emails is higher than the frequency of answering, below a critical priority, the mailbox fills up ad infinitum. We prove a theorem about the limiting shape of the mailbox just above the critical point, linking it to the convex hull of Brownian motion. We conjecture that this limiting shape is universal in a class of similar models, including a model for the evolution of an order book due to Stigler and Luckock.

Monotone and additive Markov process duality

Abstract: This paper develops a systematic treatment of monotonicity-based dualities for Markov processes taking values in partially ordered sets We show that every Markov process that takes values in a finite partially ordered set and whose generator can be represented in monotone maps has a pathwise dual, which in the special setting of attractive spin systems has been discovered earlier by Gray This dual simplifies a lot in the special case that the space is a lattice and all monotone maps satisfy an additivity condition This leads to a unified treatment of several well-known dualities, including Siegmunds dual for processes with a totally ordered state space, duality of additive spin systems, and a duality due to Krone for the two-stage contact process It is well-known that additive spin systems can be constructed using a graphical representation involving open paths We show that more generally, every additive Markov process can be formulated in terms of open paths on a suitably chosen underlying space However, in order for the process and its dual to be representable on the same underlying space, one needs to assume that the state space is a distributive lattice In the final section, we show how our results can be generalized from finite state spaces to interacting particle systems with finite local state spaces