ICTS

We introduce the theory of large deviations, beginning with Cramer's theorem and its historical significance. Particular emphasis will be placed on the analytic structures underlying the theory, especially convex duality and the Fenchel–Legendre transform, and on how these ideas arise naturally in the study of rare events. We will discuss fundamental results such as the Sanov's theorem, and elements of Donsker–Varadhan theory, highlighting the unifying role of analysis in understanding probabilistic phenomena.

We introduce the theory of large deviations, beginning with Cramer's theorem and its historical significance. Particular emphasis will be placed on the analytic structures underlying the theory, especially convex duality and the Fenchel–Legendre transform, and on how these ideas arise naturally in the study of rare events. We will discuss fundamental results such as the Sanov's theorem, and elements of Donsker–Varadhan theory, highlighting the unifying role of analysis in understanding probabilistic phenomena.

We introduce the theory of large deviations, beginning with Cramer's theorem and its historical significance. Particular emphasis will be placed on the analytic structures underlying the theory, especially convex duality and the Fenchel–Legendre transform, and on how these ideas arise naturally in the study of rare events. We will discuss fundamental results such as the Sanov's theorem, and elements of Donsker–Varadhan theory, highlighting the unifying role of analysis in understanding probabilistic phenomena.

This lecture series will explore the fundamentals of jump processes and their associated large-deviation principles as a motivation for generalized gradient structures. It will also cover the definitions and preliminary results related to generalized gradient structures, along with existence and uniqueness results for generalized gradient flows. Finally, the series will discuss applications of these concepts to reversible interacting population dynamics.

This lecture series will explore the fundamentals of jump processes and their associated large-deviation principles as a motivation for generalized gradient structures. It will also cover the definitions and preliminary results related to generalized gradient structures, along with existence and uniqueness results for generalized gradient flows. Finally, the series will discuss applications of these concepts to reversible interacting population dynamics.

This series of lectures intends to cover

(1) Introduction to the optimal transport theory. Monge and Kantorovich problems, Wasserstein distances, connections with the Monge Ampère equation.

(2) The Jordan-Kinderlehrer-Otto scheme: brief reminders about gradient flows in linear and metric spaces, Implicit Euler scheme, minimizing movements and the JKO scheme; Variants corresponding to x'=Dh*(-DF(x)) for convex h (power-like or not); the energy dissipation principle in the Wasserstein space; first properties of the JKO scheme.

(3) Convergence of the JKO scheme via the limit of the PDE (piecewise constant and piecewise geodesic interpolations, identification of the limits) and via the energy dissipation principle (the geodesically convex case via the flow interchange and the general case via the variational interpolation).

(4) Higher-order estimates and applications to the PDE (decrease of the Fisher information), to functional inequalities (proof of  the log-Sobolev inequality using the heat flow and using the JKO scheme, and generalizations), and to the strong convergence of the JKO scheme (strong L2H2 convergence via second-order estimates for the JKO scheme in the case of the Fokker-Planck equation).

This lecture series will explore the fundamentals of jump processes and their associated large-deviation principles as a motivation for generalized gradient structures. It will also cover the definitions and preliminary results related to generalized gradient structures, along with existence and uniqueness results for generalized gradient flows. Finally, the series will discuss applications of these concepts to reversible interacting population dynamics.

This series of lectures intends to cover

(1) Introduction to the optimal transport theory. Monge and Kantorovich problems, Wasserstein distances, connections with the Monge Ampère equation.

(2) The Jordan-Kinderlehrer-Otto scheme: brief reminders about gradient flows in linear and metric spaces, Implicit Euler scheme, minimizing movements and the JKO scheme; Variants corresponding to x'=Dh*(-DF(x)) for convex h (power-like or not); the energy dissipation principle in the Wasserstein space; first properties of the JKO scheme.

(3) Convergence of the JKO scheme via the limit of the PDE (piecewise constant and piecewise geodesic interpolations, identification of the limits) and via the energy dissipation principle (the geodesically convex case via the flow interchange and the general case via the variational interpolation).

(4) Higher-order estimates and applications to the PDE (decrease of the Fisher information), to functional inequalities (proof of  the log-Sobolev inequality using the heat flow and using the JKO scheme, and generalizations), and to the strong convergence of the JKO scheme (strong L2H2 convergence via second-order estimates for the JKO scheme in the case of the Fokker-Planck equation).