标题： 连续图动力学通过种群动力学：适定性、对偶性和平衡态 显示英文标题

标题： Continuum graph dynamics via population dynamics: well-posedness, duality and equilibria

摘要： 在本文中，我们考虑取值于一个称为图元的连续图集合的随机过程，图元被定义为在不可数波兰空间（具有连续统的基数）中嵌入的由N标记的顶点序列的等价类，以及一个N x N的连接矩阵，其元素为0或1，用于指定顶点对之间边的缺失或存在。 特别是，我们在一个波兰状态空间G上构造了一个马尔可夫过程，该空间适合描述可数图的时间-空间路径。 我们提出的动力学类是通过指定有限图演化的简单规则，并取无限图的极限而得到的。 图元的演化由适定的鞅问题表征，并导致具有Feller性质的强马尔可夫过程。 显示英文摘要

摘要： In this paper we consider stochastic processes taking values in a set of continuum graphs we call graphemes, defined as equivalence classes of sequences of vertices labelled by N embedded in an uncountable Polish space (with the cardinality of the continuum), together with an N x N connection matrix with entries 0 or 1 specifying the absence or presence of edges between pairs of vertices. In particular, we construct a Markov process on a Polish state space G of graphemes suitable to describe the time-space path of countable graphs. The class of dynamics we propose arises by specifying simple rules for the evolution of finite graphs and passing to the limit of infinite graphs. The evolution of graphemes is characterised by well-posed martingale problems, and leads to strong Markov processes with the Feller property.