论文标题
平均田间森林火灾年龄演化方程的良好性
Well-posedness of the mean field forest fire age evolution equation
论文作者
论文摘要
我们证明了一个微分方程的良好性，它描述了Ráth和Tóth的平均田间森林火灾模型（Arxiv：0808.2116）的经验年龄量度的大型限制的演变。这种森林火灾模型是$ N $顶点的随机图形过程，其动力学结合了Erdős-rényi动力学和闪电罢工的泊松雨。一旦通过闪电击中其任何顶点，任何连接组件中的所有边缘都会立即删除。每个顶点的年龄为年龄，以$ 1 $的价格增加，但每次燃烧时都将其重置为$ 0 $。我们考虑该模型显示自组织的关键性的渐近闪电制度。 Crane，Ráth和Yeo（Arxiv：1811.07981）以初始状态为不均匀的随机图，其边缘概率取决于顶点的年龄。他们表明，作为$ n \ to \ infty $，经验年龄分布将作为确定性自主微分方程解决方案的过程收敛。它是一个非线性年龄依赖性的人群动力学模型，其特定年龄的死亡模量涉及相关多元分支过程的分支运算符的领先征函数。微分方程在分支运算符的领先特征值$ 1 $的情况下显示出自组织的批判性，而无需将其作为边界条件。
We prove the well-posedness of a differential equation that describes the evolution of the large-system limit of the empirical age measure in the mean field forest fire model of Ráth and Tóth (arXiv:0808.2116). This forest fire model is a random graph process on $n$ vertices, whose dynamics combine the Erdős-Rényi dynamics with a Poisson rain of lightning strikes. All edges in any connected component are deleted as soon as any of its vertices is struck by lightning. Each vertex has an age, which increases at rate $1$ but is reset to $0$ each time it burns. We consider the asymptotic lightning regime in which the model displays self-organized criticality. Crane, Ráth and Yeo (arXiv:1811.07981) take the initial state to be an inhomogeneous random graph whose edge probabilities depend on the ages of the vertices. They show that as $n \to \infty$ the empirical age distribution converges as a process to the solution of a deterministic autonomous differential equation. It is a nonlinear age-dependent population dynamics model whose age-specific mortality modulus involves the leading eigenfunction of the branching operator of an associated multitype branching process. The differential equation displays self-organized criticality in the sense that the leading eigenvalue of the branching operator is held at $1$ without this being imposed as a boundary condition.
