Mixed local and nonlocal equations: analytic, numerical and probabilistic aspects

The importance of mathematical models in Science in Engineering is, by now, almost impossible to overshadow. Physics, Biology, Material Sciences, Social and Economic Sciences, or Data Science and Machine Learning are only a few of the many instances in which accurate mathematical models and efficient numerical methods to compute them are of prime importance to understand, predict and/or optimize the underlying phenomena. At the same time, questions from the applied sciences have long been a source of inspiration for deep questions within the realm of pure mathematics. In this context, our project aims to contribute to the study of the interaction between local, or integer order, and nonlocal, or fractional order, equations. Both of these separately (since the 19th century the former and more recently the latter) have received much attention from the Partial Differential Equations, Probabilistic and Numerical Analysis Communities. However, it is only in the past five years or so that these communities have directed their interest to the interaction between them. Since they account for different types of interactions, namely infinitesimal and long-range respectively, their study is both of intrinsic mathematical interest and an important contribution to the understanding of phenomena occurring in the applied sciences. We will approach the study from the analytic, probabilistic and computational/numerical points of view, as we consider that different aspects of the problem are complementary in the overall understanding of the models in question. The specific problems to be studied have connections to problems in material sciences and phase transitions, game theory, image processing, population dynamics andoptimal control theory, among others..

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