论文标题
自适应接触过程的缩放限制
Scaling limit of an adaptive contact process
论文作者
论文摘要
我们介绍并研究了一个相互作用的粒子系统，该系统在$ d $二维的圆环$（\ mathbb z/n \ mathbb z）^d $上发展。圆环的每个顶点可以被$λ\ in Type of（0，\ infty）$的个体占用。 $λ$类型的个体与费率一世，并在每个附近的空位置下生下$λ$；此外，当出生发生时，新生儿可能与父母具有相同的类型，但可能是突变体的可能性很小。 $λ$的个体的突变孩子根据概率内核选择了类型。当$ n \ to \ infty $和参数$δ_n$倾向于零快速以使突变足够及时分离时，我们考虑了此过程的渐近行为，以使具有多种类型的配置所花费的时间可忽略不计。我们表明，在适当的时间缩放和删除了多种类型的配置上所花费的时间的时间之后，该过程收敛到Markov跳跃过程，以$（0，\ infty）$（我们的费率）为$。
We introduce and study an interacting particle system evolving on the $d$-dimensional torus $(\mathbb Z/N\mathbb Z)^d$. Each vertex of the torus can be either empty or occupied by an individual of type $λ\in (0,\infty)$. An individual of type $λ$ dies with rate one and gives birth at each neighboring empty position with rate $λ$; moreover, when the birth takes place, the newborn individual is likely to have the same type as the parent, but has a small probability of being a mutant. A mutant child of an individual of type $λ$ has type chosen according to a probability kernel. We consider the asymptotic behavior of this process when $N\to \infty$ and the parameter $δ_N$ tends to zero fast enough that mutations are sufficiently separated in time, so that the amount of time spent on configurations with more than one type becomes negligible. We show that, after a suitable time scaling and deletion of the periods of time spent on configurations with more than one type, the process converges to a Markov jump process on $(0,\infty)$, whose rates we characterize.
