A first order Hamilton-Jacobi equation theory in space of probability measures, and in metric spaces

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Presentation Name：	A first order Hamilton-Jacobi equation theory in space of probability measures, and in metric spaces
Presenter🧝🏼：	Professor Jin Feng
Date：	2014-01-03
Location：	光华东主楼1801室
Abstract：	系列报告纲要： ********* Part one: Motivations and overview 1. Motivational examples : a) Optimal control of masses at the mean-field level b) Variational formulation of an irrotational compressible Euler equation c) Large deviation and a statistical mechanics conjecture by Onsager ***** Part two: Hamilton-Jacobi equation in metric space 2. The basic setup a). Basics of metric space calculus, the important role of geodesic space property, b). Action minimization and optimal control on absolutely continuous curves, c). Hamilton-Jacobi PDE on functions in metric space, d). Ekeland and Borwein-Preiss variational principles, and notion of viscosity solution 3. A viscosity solution theory a). Uniqueness through the comparison principle b). Existence through dynamic programming 4. Application to a fluid mechanics problem - part one a). An equation for observables of a variational formulation of a compressible Euler equation b). The equation is well posed. c). A different formulation at the level of space of probability measures, something about mass transport theory. **** Part three: The case of space of probability measures - I, 5. An introduction to optimal mass transportation theory a). Monge-Kantorovich problem b). The theory of Y. Brenier c). Continuity equation, an important collection of curves; and the Otto's calculus d). Gradient flows in space of probability measures 6. Application to a fluid mechanics problem - part two a). Two attempts at defining Hamiltonian (incorrect but intuitive, correct but involved) b). Definition of viscosity solution, subtleties and equivalence with the metric formulation. c). Viscosity extensions, and well-posedness of Hamilton-Jacobi equation in space of probability measures 7. Optimal control of gradient flows, models a). Euler equation with an enforced entropy dissipation mechanism b). Stochastic control of masses at the mean-fields level c). Yet another class of Hamilton-Jacobi equation in space of probability measures d). Definition of viscosity solutions -- maximum principle, e). Coercive term in control, why this class of equations cannot be modeled by the previous metric space ones. ***** Part four: The case of space of probability measures - II, 8. More on optimal mass transportations, refined techniques a). A rigorous mass transportation calculus by Ambrosio-Gigli-Savare b). Log Sobolev, refined log-Sobolve and other related mass transportation inequalities. c). The case of controlled gradient flows, free energy as a Lyapunov functional d). Definition of viscosity solution for a Hamilton-Jacobi equation in space of probability measures - the subtle roles of test functions. e). An inequality for the comparison of Hamiltonians 9. Optimal control of gradient flows, a Hamilton-Jacobi PDE theory a). Viscosity solution, definition and inequalities revisited b). A proof of the comparison principle ****** Part five 10. An introduction to a theory of large deviation for metric space valued Markov process a). Martingale problems b). Variational problems in the context of large deviation c). Convergence of Hamiltonians implies large deviation d). The important role of comparison principle
Annual Speech Directory：	No.2

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