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Edwin A. Perkins

Publications and source records attributed to Edwin A. Perkins.

10 recordsLinked to original sources
arXiv2019

Rescaling the spatial lambda Fleming-Viot process and convergence to super-Brownian motion

We show that a space-time rescaling of the spatial Lamba-Fleming-Viot process of Barton and Etheridge converges to super-Brownian motion. This can be viewed as an extension of a result of Chetwynd-Diggle and Etheridge (2018). In that work the scaled impact factors (which govern the event based dynamics) vanish in the limit, here we drop that requirement. The analysis is particularly interesting in the biologically relevant two-dimensional case.

math.PR↗
arXiv2014

A complete convergence theorem for voter model perturbations

We prove a complete convergence theorem for a class of symmetric voter model perturbations with annihilating duals. A special case of interest covered by our results is the stochastic spatial Lotka-Volterra model introduced by Neuhauser and Pacala [Ann. Appl. Probab. 9 (1999) 1226-1259]. We also treat two additional models, the "affine" and "geometric" voter models.

math.PR↗
arXiv2014

A phase transition for measure-valued SIR epidemic processes

We consider measure-valued processes $X=(X_t)$ that solve the following martingale problem: for a given initial measure $X_0$, and for all smooth, compactly supported test functions $φ$, \begin{eqnarray*}X_t(φ)=X_0(φ)+\frac{1}{2}\int _0^tX_s(Δφ)\,ds+θ\int_0^tX_s(φ)\,ds\\{}-\int_0^tX_s(L_sφ)\,ds+M_t(φ).\end{eqnarray*} Here $L_s(x)$ is the local time density process associated with $X$, and $M_t(φ)$ is a martingale with quadratic variation $[M(φ)]_t=\int_0^tX_s(φ^2)\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $θ_c(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $θ>θ_c(d)$, then the solution survives forever with positive probability, but if $θ<θ_c(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $θ$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $θ$; that is, for any compact set $K$, the process $X_t(K)=0$ eventually.

math.PR↗
arXiv2012

Uniqueness in law for parabolic SPDEs and infinite-dimensional SDEs

We prove uniqueness in law for a class of parabolic stochastic partial differential equations in an interval driven by a functional A(u) of the temperature u times a space-time white noise. The functional A(u) is Hölder continuous in u of order greater than 1/2. Our method involves looking at an associated system of infinite-dimensional stochastic differential equations and we obtain a uniqueness result for such systems.

math.PR↗
arXiv2008

Degenerate stochastic differential equations arising from catalytic branching networks

We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case: $$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2γ_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ where $X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $γ$ and $b$ are continuous on the positive orthant, $γ$ is strictly positive, and $b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i=θ_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times,and a refined integration by parts technique from Dawson-Perkins]. As a by-product of the proof we obtain the strong Feller property of the associated resolvent.

math.PR↗
arXiv2005

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.

math.PR↗
arXiv2005

Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type

We consider the operator $$\sL f(x)=\tfrac12 \sum_{i,j=1}^\infty a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i b_i(x) \frac{\del f}{\del x_i}(x).$$ We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the $a_{ij}, b_i$, and $\lam_i$. The process corresponding to $\sL$ solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein-Uhlenbeck process.

math.PR↗
arXiv2004

Diffusion limited aggregation on a tree

We study the following growth model on a regular d-ary tree. Points at distance n adjacent to the existing subtree are added with probabilities proportional to alpha^{-n}, where alpha<1 is a positive real parameter. The heights of these clusters are shown to increase linearly with their total size; this complements known results that show the height increases only logarithmically when alpha>=1. Results are obtained using stochastic monotonicity and regeneration results which may be of independent interest. Our motivation comes from two other ways in which the model may be viewed: as a problem in first-passage percolation, and as a version of diffusion-limited aggregation (DLA), adjusted so that `fingering' occurs.

math.PR↗