Lecture courses
- Prof. Franco Flandoli (Scuola Normale Superiore di Pisa)
Lecture: An introduction to random dynamical systems for climate
Short biography: He is currently a full professor in Probability and Statistics, Scuola Normale Superiore of Pisa. His main research interests include Stochastic ordinary and partial differential equations Stochastic models in fluid dynamics Random dynamical systems Macroscopic limits of particle systems.
Personal website: http://users.dma.unipi.it/flandoli/
Material of the course: download - Prof. Dan Crisan (Imperial College London)
Lecture: Particle filters for data assimilation
Short biography: He is currently a professor of Mathematics at the Department of Mathematics of Imperial College London and Director of the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth http://mpecdt.org/. His long-term research interests lie broadly in Stochastic Analysis, a branch of Mathematics that looks at understanding and modelling systems that behave randomly.
Personal website: https://www.ma.ic.ac.uk/~dcrisan/
Material of the course: download
Invited lectures
- Dr. Thomas Cass (Imperial College London)
Lecture: Some old and new results on the signature transform of rough path theory
Abstract: The work of Lyons (1998) introduced the general theory of rough paths and rough differential equations. A central object is the (path) signature, a non-commutative power series of iterated integrals . The seminal paper of Hambly and Lyons (2010) built upon the earlier geometric work of K-T Chen to develop the modern mathematical foundations of the theory of the signature. More recently, the signature has been used as a feature set for problems in data science.
We survey some of the mathematical underpinnings of this theory and illustrate its use through a range of recent results and applications.
Personal website: https://www.imperial.ac.uk/people/thomas.cass - Prof. Benjamin Gess (Max Planck Institute for Mathematics in the Sciences and University of Bielefeld)
Lecture: Stochastic PDE, non-equilibrium fluctuations and large deviations
Abstract: Macroscopic fluctuation theory provides a general framework for far from equilibrium thermodynamics, based on a fundamental formula for large fluctuations around (local) equilibria. This fundamental postulate can be informally justified from the framework of fluctuating hydrodynamics, linking far from equilibrium behavior to zero-noise large deviations in conservative, stochastic PDE. In this talk, we will give rigorous justification to this relation in the special case of the zero range process. More precisely, we show that the rate function describing its large fluctuations is identical to the rate function appearing in zero noise large deviations to conservative stochastic PDE. The proof is based on the well-posedness of the skeleton equation -- a degenerate parabolic-hyperbolic PDE with irregular coefficients, the proof of which extends DiPerna-Lions' concept of renormalized solutions to nonlinear diffusions.
Personal website: http://www.bgess.de/.
Slide of Presentation: download - Dr. Hoang Duc Luu (Institute of Mathematics, VAST and Max Planck Institute for Mathematics in the Sciences)
Lecture: Asymptotic stability and stationary states for stochastic systems: a pathwise approach
Abstract: Since the pioneer work by Lyapunov, stability theory for stochastic systems has been an interesting object of mathematical research, often motivated by many applications. Under the influence of standard Brownian noises, a traditional approach assumes the existence of a Lyapunov-type function and apply the Ito’s formula to confirm the exponential stability of the equilibrium, hence the system is exponentially stable in the mean-square sense, which implies the exponential stability in the path-wise sense. Another geometric approach considers the Fokker Planck equation and the generated Markov semigroup, then proves the ergodicity of the unique stationary distribution by combining tools in the Γ−calculus, the logarithmic Sobolev inequality, and the curvature-dimension condition to obtain exponential rate of convergence in the Kullback-Leibler divergence.
However when the driving noise is neither Markov nor semi-martingale (e.g. fractional Brownian motions), less is known on the asymptotic stability. Such systems, often called rough differential/evolution equations, can be solved either with Lyons’ theory of rough paths, in particular the rough integrals are understood in the Gubinelli sense for controlled rough paths, or with fractional calculus. When the noises are fractional Brownian motions, there are also Hairer's works on ergodicity of the unique stationary distribution that attracts others in the total variation norm.
In this talk, I will present an analytic approach to study the long term behavior of rough equations and the stochastic stability of its stationary states. Using the framework of random dynamical systems and random attractors, one can prove the existence and upper semi-continuity of a global pullback attractor. In particular, the techniques generalize two classical Lyapunov methods in proving exponential stability.
Personal website: https://www.mis.mpg.de/jjost/members/hoang-duc-luu.html.
Slide of Presentation: download - Prof. Juan-Pablo Ortega (Nanyang Technological University)
Lecture: Reservoir Computing and the Learning of Dynamic Processes
Abstract: Dynamic processes regulate the behaviour of virtually any artificial and biological agent, from stock markets to epidemics, from driverless cars to healthcare robots. The problem of modeling, forecasting, and generally speaking learning dynamic processes is one of the most classical, sophisticated, and strategically significant problems in the natural and the social sciences. In this talk we shall discuss both classical and recent results on the modeling and learning of dynamical systems and input/output systems using an approach generically known as reservoir computing. This information processing framework is characterized by the use of cheap-to-train randomly generated state-space systems for which promising high-performance physical realizations with dedicated hardware have been proposed in recent years. In our presentation we shall put a special emphasis in the approximation properties of these constructions.
Personal website: https://juan-pablo-ortega.com/
Slide of Presentation: download - Prof. Dong Zhao (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Lecture: Large time behavior of strong solutions for stochastic Burgers equation
Abstract:
Personal website:
Slide of Presentation: download