Edwin Perkins

Collision local times and measure-valued processes

We consider two independent Dawson-Watanabe super-Brownian motions, Y and Y. These processes are diffusions taking values in the space of finite measures on R . We show that if d < 5 then with positive probability there exist times / such that the closed supports of Yj and Y intersect; whereas if d > 5 then no such intersections occur. For the case d < 5, we construct a continuous, non-decreasing measure-valued process L(Y, Y), the collision local time, such that the measure defined by [0, t] x B i—• U(Y, Y)(B), B £ #(R ), is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition of L(Y, Y). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions with point interactions. In the course of our proofs we obtain smoothness results for the random measures Yt that are uniform in t. These theorems use a nonstandard description of Y and are of independent interest.

Rescaled Lotka-Volterra Models Converge to Super-Brownian Motion

We show that a sequence of stochastic spatial Lotka-Volterra models, suitably rescaled in space and time, converges weakly to super-Brownian motion with drift. The result includes both long range and nearest neighbor models, the latter for dimensions three and above. These theorems are special cases of a general convergence theorem for perturbations of the voter model.

MUTUALLY CATALYTIC BRANCHING IN THE PLANE: UNIQUENESS

Abstract We study a pair of populations in R 2 which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. Previous work had established the existence of such a process and derived some of its small scale and large scale properties. This paper is primarily focused on the proof of uniqueness of solutions to the martingale problem associated with the model. The self-duality property of solutions, which is crucial for proving uniqueness and was used in the previous work to derive many of the qualitative properties of the process, is also established.

Rescaled Particle Systems Converging to Super-Brownian Motion

Super-Brownian motion was originally constructed as a scaling limit of branching random walk. Here we describe recent results which show that, in two or more dimensions, it is also the limit of long range contact processes and long, short, and medium range voter models.

Renormalization of the two-dimensional Lotka–Volterra model

We show that renormalized two-dimensional Lotka--Volterra models near criticality converge to a super-Brownian motion. This is used to establish long-term survival of a rare type for a range of parameter values near the voter model.

Extinction for two parabolic stochastic PDE's on the lattice

It is well known that, starting with finite mass, the super-Brownian motion dies out in finite time. The goal of this article is to show that with some additional work, one can show finite time die-out for two types of systems of stochastic differential equations on the lattice Zd.rnrnFor our first system, let 1/2⩽γ<1, and consider non-negative solutions of rnrndu(t,x)=Δu(t,x)dt+uγ(t,x)dBx(t),x∈Zd, rnrnu(0,x)=u0(x)⩾0.rnrnHere Δ is the discrete Laplacian and {Bx:x∈Zd} is a system of independent Brownian motions. We assume that u0 has finite support. When γ=1/2, the measure which puts mass u(t,x) at x is a super-random walk and it is well-known that the process becomes extinct in finite time a.s. Finite-time extinction is known to be a.s. false if γ=1. For 1/2<γ<1, we show finite-time die-out by breaking up the solution into pieces, and showing that each piece dies in finite time. Unlike the superprocess case, these pieces will not in general evolve independently.rnrnOur second example involves the mutually catalytic branching system of stochastic differential equations on Zd, which was first studied in Dawson and Perkins (1998). rnrndUt(x)=ΔUt(x)dt+Ut(x)Vt(x)dB1,x(t), rnrndVt(x)=ΔVt(x)dt+Ut(x)Vt(x)dB2,x(t), rnrnU0(x)⩾0, rnrnV0(x)⩾0.rnrnBy using a somewhat different argument, we show that, depending on the initial conditions, finite time extinction of one type may occur with probability 0, or with probability arbitrarily close to 1.

Markov Processes, Characterization and Convergence.

Introduction. 1. Operator Semigroups. 2. Stochastic Processes and Martingales. 3. Convergence of Probability Measures. 4. Generators and Markov Processes. 5. Stochastic Integral Equations. 6. Random Time Changes. 7. Invariance Principles and Diffusion Approximations. 8. Examples of Generators. 9. Branching Processes. 10. Genetic Models. 11. Density Dependent Population Processes. 12. Random Evolutions. Appendixes. References. Index. Flowchart.