Generalising Nonlinear Population Models - Radon Measures, Polish Spaces and the Flat Norm

In this work, we explore Radon measures and their role in extending nonlinear structured population models from the classical Euclidean setting to abstract Polish metric spaces. We begin by studying the flat norm on the space of Radon measures, and show how it generalises the Wasserstein distance W_1 from conservative to unbalanced cases.

In the second part, we prove well-posedness for structured population models formulated in measures in the Euclidean setting and then extend these results to Polish metric spaces. However, in the absence of a vector space structure, we cannot rely on a governing differential equation, and thus have to use an implicit integral representation instead. This approach leverages the favorable functional analytic properties of the space of measures under the flat norm.

Finally, we outline potential applications of this framework, including measure differential equations and coagulation-fragmentation models, and demonstrate its connection to transport distances used in nonexpanding transport processes.