以个人偏好发展社区
Evolving Communities with Individual Preferences
摘要：
本文的目标是提供数学上严格的工具，模拟社区的相互作用的个人的演变。我们通过一个测度空间来对总体进行建模，其中度量决定了个体偏好的丰富程度。个人的偏好被描述为一个粗糙路径的可测量的选择。我们专注于弱相互作用系统的情况下，在那里我们能够展示一致的解决方案的存在性和唯一性。一般来说，解是相互作用的线的连续体，类似于共同构成流体运动的大量单个原子轨迹。种群的演化不需要由任何过度偏微分方程来控制。虽然在这种情况下，可以将McKean-Vlasov型的标准非线性抛物型偏微分方程与社区的具体例子相匹配。进化种群的整体行为提供了偏微分方程的解。一个重要的技术成果是系统行为的连续性，相对于分配权重给个人的措施的变化。用独立同分布的经验分布代替确定性测度。从它的样本导致许多标准的模型，并应用连续性的结果可以很容易地证明混沌的传播。这里提出的严格的基础导致简单的模型，在物理和社会科学中具有广泛的适用性。我们并不假定宏观动力学可以用偏微分方程来模拟。这项工作的基础上，考虑到系统的极限行为，大量的粒子相互作用的独立偏好的概率文献;也有工作的连续模型的偏好描述的半鞅测度。我们提到一些关键文件。
英文摘要：
The goal of this paper is to provide mathematically rigorous tools for modelling the evolution of a community of interacting individuals. We model the population by a measure space where the measure determines the abundance of individual preferences. The preferences of an individual are described by a measurable choice of a rough path. We focus on the case of weakly interacting systems, where we are able to exhibit the existence and uniqueness of consistent solutions. In general, solutions are continuum of interacting threads analogous to the huge number of individual atomic trajectories that together make up the motion of a fluid. The evolution of the population need not be governed by any over-arching PDE. Although one can match the standard nonlinear parabolic PDEs of McKean-Vlasov type with specific examples of communities in this case. The bulk behaviour of the evolving population provides a solution to the PDE. An important technical result is continuity of the behaviour of the system with respect to changes in the measure assigning weight to individuals. Replacing the deterministic measure with the empirical distribution of an i.i.d. sample from it leads to many standard models, and applying the continuity result allows easy proofs for propagation of chaos. The rigorous underpinning presented here leads to uncomplicated models which have wide applicability in both the physical and social sciences. We make no presumption that the macroscopic dynamics are modelled by a PDE. This work builds on the fine probability literature considering the limit behaviour for systems where a large no of particles are interacting with independent preferences; there is also work on continuum models with preferences described by a semi-martingale measure. We mention some of the key papers.
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