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Mean field systems on networks, with singular interaction through hitting times. (English) Zbl 1448.82032
The article belongs to a group of works dealing with the investigation of the mean field particle sytems with singular interaction through hitting times. The second section contains a detailed description of the proposed particle systems and its dynamics. The system of particles is characterized by their “level of healthiness” \(Y^x\), taking values in \([0,\infty)\) and by their “type” \(x\in\mathcal{X}\), \(\mathcal{X}\) an abstract finite set. These “level of healthiness “ are defined as a family of sochastic processes with a dynamics of the form \({Y_t}^x = {Z_t}^x + C(x) \int_{\mathcal{X}}g({\mathbb{P}}({\tau}^{x'}>t)){\kappa}(x,dx')\). The interaction function \(g\) is assumed continuous and nondecreasing on the interval \((0,1]\).
As practical interpretation of the system, the particles may represent cells, individuals or organisations with exposures to each other, for instance a neuron in the human brain is physically connected to neighbouring neurons or a computer can be part of a network. Under a special assumption the existence of a solution to the considered model is proved. The uniqueness of the solution is not part of the investigation. In the third section the times of fragility of the considered particle systems are studied. Necessary and sufficient conditions for fragility are proved. The forth section is devoted to the study of controlled mean field dynamics and equilibrium. In the considered game, every particle is allowed to control the strength of its connections dynamically, however by optimizing its objective. The optimization problem and the notion of equilibrium are introduced, one proves the existence of an equilibrium and one developes a technique to its construction. In the second section a generalization of Schauder’s fixed-point theorem for mappings defined on the Skorohod space with the M1 tolopogy is proved. Proofs and complementary results are contained in Appendix A and Appendix B.

82C22 Interacting particle systems in time-dependent statistical mechanics
05C57 Games on graphs (graph-theoretic aspects)
54H25 Fixed-point and coincidence theorems (topological aspects)
92C20 Neural biology
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