Basic theory of measure-based models

This chapter is devoted to the mathematical foundations of the model introduced in Chap.  5. Contents go continuously back and forth between modeling and analysis, however with a more formal approach than that used in the previous chapter. The first three sections, from Sects. 6.1 to 6.3, discuss how the measure-based model can be derived from a particle description of pedestrians, thereby formalizing the link between individualities and collectivity which is at the basis of most of the complexity of crowd behaviors. In addition, in the light of such a derivation they propose a probabilistic reading of the measure-based model, which turns out to be particularly meaningful for applications. The central part of the chapter, encompassing Sects. 6.4–6.7, is concerned with the basic theory of well-posedness and numerical approximation of measure-valued Cauchy problems for first order models based on conservation laws, also in a multiscale perspective. Minimal generic assumptions are stated in order to achieve proofs, to be regarded possibly also as guidelines in the modeling approach. Finally, Sect. 6.8 resumes the discussion about the crowd model presented in Chap.  5 studying under which conditions it is in the scope of the theory set forth in the preceding sections.