Abstract
One can construct a sequence of rescaled perturbations of voter processes in dimension $d=1$ whose approximate densities are tight. By combining both long-range models and fixed kernel models in the perturbations and considering the critical long-range case, results of Cox and Perkins (2005) are refined. As a special case we are able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition. In the case of long-range interactions only, the approximate densities converge to continuous space time densities which solve a class of SPDEs (stochastic partial differential equations), namely the heat equation with a class of drifts, driven by Fisher-Wright noise. If the initial condition of the limiting SPDE is integrable, weak uniqueness of the limits follows. The results obtained extend the results of Mueller and Tribe (1995) for the voter model by including perturbations. In particular, spatial versions of the Lotka-Volterra model as introduced in Neuhauser and Pacala (1999) are covered for parameters approaching one. Their model incorporates a fecundity parameter and models both intra- and interspecific competition.
Highlights
We define a sequence of rescaled competing species models ξNt in dimension d = 1, which can be described as perturbations of voter models
We label the state of site x at time t by ξNt (x) where ξNt (x) = 0 if the site is occupied at time t by type 0 and ξNt (x) = 1 if it is occupied by type 1
In [13], Mueller and Tribe show that the approximate densities of type 1 of rescaled biased voter processes converge to continuous space time densities which solve the heat equation with drift, driven by Fisher-Wright noise
Summary
Following Definition VI.3.25 in Jacod and Shiryaev [8], we shall say that a collection of processes with paths in D(S) is C-tight if and only if it is tight in D(S) and all weak limit points are a.s. continuous. In what follows we shall investigate tightness of {A(ξ·N ) : N ≥ 1} in D( 1) and tightness of {νtN : N ≥ 1} in D( (R)), where (R) is the space of Radon measures equipped with the vague topology ( (R) is Polish, see Kallenberg [9], Theorem A2.3(i)). On we consider the special case of no fixed kernel interaction ( to be called no shortrange competition in what follows, for reasons that become clear later) and investigate the limits of our tight sequences. If we assume 〈u0, 1〉 < ∞, ut is the unique in law [0, 1]valued solution to the above SPDE
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