Speakers and Topics

We cover the core functional-analytic tools underlying modern PDE theory and gradient flows. We begin with classical Lebesgue and Sobolev spaces and their embedding theorems, then discuss weak and strong convergence -- including weak-* compactness and Bochner spaces for time-dependent problems. We next recall the notion of weak solutions of elliptic and parabolic PDEs. The series concludes with key functional inequalities, the Poincaré and log-Sobolev inequalities, and their connection to spectral gaps. Together, these topics provide the essential background for engaging with advanced material on gradient flows and related areas.

Data assimilation refers to the problem of estimation of the state of a high dimensional chaotic system given noisy, partial observations and a dynamical model of the system. These talks will focus on the Bayesian viewpoint of data assimilation and connections with inverse problems. After introducing the problem using hidden Markov models, the talks will discuss the most commonly used algorithms, namely, ensemble Kalman filter (EnKF) and particle filter (PF). The lectures will be devoted to discussing some theoretical problems as well as their relation to the numerical implementation of the EnKF and PF.

Inverse problems: A Bayesian perspective, by A. M. Stuart, Acta Numerica. 2010;19:451-559 https://doi.org/10.1017/S0962492910000061

Inverse Problems and Data Assimilation (chapter 13)

Daniel Sanz-Alonso, Andrew M. Stuart, Armeen Taeb

https://www.cambridge.org/core/books/inverse-problems-and-data-assimilation/CD12492AFB6C17FA0440813C8FD445A3

https://arxiv.org/abs/1810.06191

We study the long-time behavior of solutions to the Fokker-Planck equation, focusing on convergence to equilibrium. Working in a weighted L2 framework, we show how the Poincaré inequality yields quantitative decay estimates toward the stationary state. We then introduce Lyapunov functionals as a complementary and more general tool for capturing convergence rates in dissipative systems, highlighting the interplay between functional inequalities and entropy-type methods. Together, these approaches illustrate two classical yet powerful strategies for quantifying how fast a system relaxes to equilibrium.

We explore discrete approaches to evolution equations and optimal transport. We begin by constructing weak solutions to evolution equations via time discretization, focusing on the implicit Euler scheme as a variational tool for approximating gradient flows. We then turn to the geometric structure underlying discretized optimal transport, examining the inverse distribution function in one dimension and its multi-dimensional analogue, Laguerre cells, as natural discrete objects encoding optimal transport maps. Together, these topics highlight how discretization (in time and in space) provides both a computational and conceptual bridge to the continuum theory of gradient flows.

We introduce the foundations of optimal transport and its associated geometry. We begin with the Monge–Kantorovich duality, framed through the lens of linear programming, before covering the basics of optimal transport: its primal formulation, dual problem, and geometric interpretation via the Brenier map. We then construct the Wasserstein metric from optimal transport costs, focusing on the W2​ distance, its metric and topological properties, geodesics in Wasserstein space, and its connection to the Monge–Ampère equation. The series concludes with the study of geodesically convex functionals on metric spaces, introducing the notion of metric derivative and discussing consequences such as uniqueness of minimizers. Together, these topics lay the geometric and variational groundwork for the theory of gradient flows in the Wasserstein space.

We develop the theory of continuous-time stochastic processes, building toward the study of large deviations for jump processes. We begin with continuous-time Markov chains, covering transition rates, the generator, Itô integration, the Kolmogorov forward and backward equations, stationary distributions, and detailed balance and reversibility. This serves as a natural stepping stone to jump processes more generally, including Poisson and compound Poisson processes, their generators, and the martingale problem formulation. We conclude by introducing the Skorokhod space, weak convergence of stochastic processes, and the notion of a path-space measure, laying the groundwork needed to formulate large deviation principles for jump processes.

We introduce gradient flows in finite dimensions, viewing ODEs through the lens of Lyapunov functions and energy dissipation. We study long-time behavior and convergence to equilibrium, then examine the implicit Euler scheme in R^n as a variational warm-up to the JKO scheme in Wasserstein space.

This mini-course explores the deep and elegant interplay between analysis and probability, beginning with the notion of weak convergence of random variables and culminating in the theory of large deviations. We start with foundational material from measure theory, including push-forward measures, weak convergence, tightness, and Prokhorov’s theorem, and then develop key probabilistic concepts such as random variables, conditional expectation, martingales, and random walks. These ideas lead naturally to Donsker’s theorem and the emergence of Brownian motion as a universal scaling limit.

Building on this framework, we introduce the theory of large deviations, motivated by Cramer’s theorem and its historical significance. Particular emphasis is placed on the analytic structures underlying the theory, especially convex duality and the Fenchel-Legendre transform, and how they arise naturally in the study of rare events. We will discuss central results such as the Gartner-Ellis theorem, Sanov’s theorem, and elements of Donsker-Varadhan theory, highlighting the unifying role of analysis in the investigation of probabilistic phenomena.

References:

Ref #1: Probability and Measure by P. Billingsley

Ref #2: Probability & Measure Theory by R. B. Ash and C. A. Doleans-Dade

Ref #3: Convergence of Probability Measures by P. Billingsley

Ref #4: Probability Measures on Metric Spaces by K. R. Parthasarathy

Ref #5: Probability by L. Breiman

Ref #6: Brownian Motion by P. Morters and Y. Peres

Ref #7: Probability with Martingales by D. Williams

Ref #8: Large Deviations Techniques and Applications by A. Dembo and O. Zeitouni

We introduce the McKean-Vlasov SDE as well as corresponding PDE to describe the Microscopic as well as Macroscopic behaviour of interacting particle systems. We will prove the existence and uniqueness of a McKean-Vlasov-type SDE  by using a fixed-point argument on the space of probability measures, typically utilizing the Wasserstein metric. We then study, as the number of particles N approaches infinity, how this limit converges to an non-linear equation. The derivation of the Mean-Field Limit formalizes how a microscopic system of N interacting particles converges to a macroscopic, continuous equation (the McKean–Vlasov equation) as N goes to infinity. The core strategy is to introduce an idealized system of completely independent processes and show that the actual interacting particles stay close to these independent trajectories as N grows.

We cover the variational and convex-analytic foundations underlying modern gradient flow theory. We begin with convex analysis and duality, including subdifferentials, the Legendre–Fenchel transform, and Fenchel–Moreau duality. We then study the direct method in the calculus of variations, emphasizing lower-semicontinuity, coercivity, and convexity in proving existence of minimizers. Next, we introduce Γ-convergence, covering its definition, basic properties, and compactness results. The series concludes with gradient flows in Hilbert spaces, including the Cauchy–Lipschitz theorem, subdifferential flows, the implicit Euler scheme, and the theory of maximal monotone operators.