Mean-field equations, information inequalities and concentration bounds

The PhD project focuses on the analysis of mean-field equations in its variety of aspects and applications in Finance, Physics and Machine Learning. We develop new results on so-called concentration bounds for stochastic approximations of Euler type for the interacting particle systems used to approximate McKeanVlasov/mean-field equations. We aim at new well-posedness of mean-field type equations in its varying applications. We develop an information-theoretic framework for quantification and mitigation of error in trajectory-based predictions which are obtained from uncertain vector fields generating the underlying stochastic dynamical system on mean-field type. This is motivated by the necessity to improve predictions in multi-scale systems based on simplified, data-driven models. Here, the distance between two probability measures associated with the true dynamics and its approximation is defined via so-called phi-divergences. The goal is to
obtain general information bounds on the uncertainty in estimates of observables based on the approximate dynamics in terms of the phi-divergences. This new framework provides a systematic link between field-based model error and the resulting uncertainty in trajectory-based predictions. We seek to better understand and develop the theory of Wasserstein gradient flows and its generalization to non-dissipative systems. Results include new regularity statements for the associated mean-field equations and their approximating particle systems in association with underlying random dynamical systems.