Treffer: A journey in the fields of PDE, probabilities and statistics with point processes ; Voyage aux pays des EDP, probabilités et statistiques avec des processus ponctuels

Point processes form a family of mathematical models commonly used to model brief events (e.g. stock market orders, infections, action potentials) or the location of objects (e.g. trees, galaxies, material cracks). These models are used, from a probabilistic point of view, to explain qualitatively certain phenomena observed in practical temporal or spatial configurations, or, from a statistical point of view, to quantitatively estimate model parameters based on observed configurations. This manuscript is partly within the framework detailed above, but also deals with the link that these models have with deterministic models: differential equations and partial differential equations. Indeed, in a large-scale framework, which is particularly interesting Indeed, in a large-scale framework, which is particularly interesting for modelling a large number of interacting neurons, mean-field and diffusive approximation theories allow this link to be formalised. Thus, the three main chapters of this manuscript (Chapters II to IV) correspond to these three major areas of mathematics: Partial differential equations, Probability, Statistics.On the one hand, the preferred application framework for temporal point processes is that ofneuroscience: each dimension of the process represents a neuron and each point representsan action potential. Existence and uniqueness results for deterministic models aredetailed in Chapter II and then used on point processes in Chapter III to rigorously demonstrate the link between these two domains. From a statistical point of view, the estimation of interactions between the dimensions of the process is studied in Chapter IV.On the other hand, there is no preferred framework for spatial point processes, but a new model based on critical points of a Gaussian random field is studied. From a probabilistic point of view, the simulation of such processes is studied in Chapter III. From a statistical point of view, the link between the characteristics of the point process and those of the underlying ...