Jan M. Swart

Branching-coalescing particle systems

Abstract.We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.

Voter models with heterozygosity selection.

This paper studies variations of the usual voter model that favor types that are locally less common. Such models are dual to certain systems of branching annihilating random walks that are parity preserving. For both the voter models and their dual branching annihilating systems we determine all homogeneous invariant laws, and we study convergence to these laws started from other initial laws.

Trimmed trees and embedded particle systems

In a supercritical branching particle system, the trimmed tree consists of those particles which have descendants at all times. We develop this concept in the superprocess setting. For a class of continuous superprocesses with Feller underlying motion on compact spaces, we identify the trimmed tree, which turns out to be a binary splitting particle system with a new underlying motion that is a compensated h-transform of the old one. We show how trimmed trees may be estimated from above by embedded binary branching particle systems.

Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Abstract. Let K⊂ℝd (d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in KΛ, equipped with the product topology, where each coordinate solves a SDE of the form dXi(t) = ∑ja(j−i) (Xj(t) −Xi(t))dt + σ (Xi(t))dBi(t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel aS(i) = a(i) + a(−i) is recurrent, then each component Xi(∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if aS is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of the components. This distribution turns out to be universal in all recurrent kernels aS on Abelian groups Λ.

Pathwise uniqueness for a SDE with non-Lipschitz coefficients

We consider the ordinary stochastic differential equation on the closed unit ball E in . While it is easy to prove existence and distribution uniqueness for solutions of this SDE for each c[greater-or-equal, slanted]0, pathwise uniqueness can be proved by standard methods only in dimension n=1 and in dimensions n[greater-or-equal, slanted]2 if c=0 or if c[greater-or-equal, slanted]2 and the initial condition is in the interior of E. We sharpen these results by proving pathwise uniqueness for c[greater-or-equal, slanted]1. More precisely, we show that for X1,X2 solutions relative to the same Brownian motion, the function is almost surely nonincreasing. Whether or not pathwise uniqueness holds in dimensions n[greater-or-equal, slanted]2 for 0

Entropy-Driven Phase Transition in Low-Temperature Antiferromagnetic Potts Models

We prove the existence of long-range order at sufficiently low temperatures, including zero temperature, for the three-state Potts antiferromagnet on a class of quasi-transitive plane quadrangulations, including the diced lattice. More precisely, we show the existence of (at least) three infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one sublattice have a higher probability to be in one state than in either of the other two states. For the special case of the diced lattice, we give a good rigorous lower bound on this probability, based on computer-assisted calculations that are not available for the other lattices.

Extinction versus exponential growth in a supercritical super-Wright-Fisher diffusion

We study mild solutions u to the semilinear Cauchy problem with x[set membership, variant][0,1], f a nonnegative measurable function and [gamma] a positive constant. Solutions to this equation are given by , where is the log-Laplace semigroup of a supercritical superprocess taking values in the finite measures on [0,1], whose underlying motion is the Wright-Fisher diffusion. We establish a dichotomy in the long-time behavior of this superprocess. For [gamma][less-than-or-equals, slant]1, the mass in the interior (0,1) dies out after a finite random time, while for [gamma]>1, the mass in (0,1) grows exponentially as time tends to infinity with positive probability. In the case of exponential growth, the mass in (0,1) grows exponentially with rate [gamma]-1 and is approximately uniformly distributed over (0,1). We apply these results to show that has precisely four fixed points when [gamma][less-than-or-equals, slant]1 and five fixed points when [gamma]>1, and determine their domains of attraction.

Numerical analysis of the rebellious voter model

The rebellious voter model, introduced by Sturm and Swart (2008), is a variation of the standard, one-dimensional voter model, in which types that are locally in the minority have an advantage. It is related, both through duality and through the evolution of its interfaces, to a system of branching annihilating random walks that is believed to belong to the ‘parity-conservation’ universality class. This paper presents numerical data for the rebellious voter model and for a closely related one-sided version of the model. Both models appear to exhibit a phase transition between noncoexistence and coexistence as the advantage for minority types is increased. For the one-sided model (but not for the original, two-sided rebellious voter model), it appears that the critical point is exactly a half and two important functions of the process are given by simple, explicit formulas, a fact for which we have no explanation.

A conditional product measure theorem

Under a suitable regularity assumption, a generalisation of the product measure theorem to the case of conditional independence is given. A counterexample is constructed to show that some assumption (e.g. regularity) is needed

Rigorous results for the Stigler–Luckock model for the evolution of an order book

In 1964, G.J. Stigler introduced a stochastic model for the evolution of an order book on a stock market. This model was independently rediscovered and generalized by H. Luckock in 2003. In his formulation, traders place buy and sell limit orders of unit size according to independent Poisson processes with possibly different intensities. Newly arriving buy (sell) orders are either immediately matched to the best available matching sell (buy) order or stay in the order book until a matching order arrives. Assuming stationarity, Luckock showed that the distribution functions of the best buy and sell order in the order book solve a differential equation, from which he was able to calculate the position of two prices J − < J+ such that buy orders below J− and sell orders above J+ stay in the order book forever while all other orders are eventually matched. We extend Luckock’s model by adding market orders, i.e., with a certain rate traders arrive at the market that take the best available buy or sell offer in the order book, if there is one, and do nothing otherwise. We give necessary and sufficient conditions for such an extended model to be positive recurrent and show how these conditions are related to the prices J − and J+ of Luckock.