Lecture Series in Probability Theory
Jump Markov models and transition state theory:
the quasi-stationary distribution approach
2021. 8. 25-27
Zoom 937 905 8748
Invited Speakers:
Tony Lelièvre (Ecole des Ponts ParisTech and Inria)
Dorian Le Peutrec (Université d'Orléans)
Boris Nectoux (Université Clermont Auvergne)
Program Schedule
Lecture 1.
- Part 1: Aug 25th (Wed), 09:00-11:00 in France and 16:00-18:00 in Korea
- Part 2: Aug 26th (Thu), 08:00-09:00 in France and 15:00-16:00 in Korea
Speaker: Tony Lelièvre
Title: Introduction to the quasi-stationary distribution approach for the exit problem: theoretical and numerical aspects
Link to slide, Link to lecture-part 1, Link to lecture-part 2
Lecture 2.
- Part 1: Aug 26th (Thu), 09:00-11:00 in France and 16:00-18:00 in Korea
- Part 2: Sep 1st (Wed), 09:00-10:00 in France and 16:00-17:00 in Korea
Speaker: Dorian Le Peutrec
Title: Semi-classical analysis and Witten Laplacians
Link to slide, Link to lecture-part 1, Link to lecture-part 2
Lecture 3.
Aug 27th (Fri), 09:00-11:00 in France and 16:00-18:00 in Korea
Speaker: Boris Nectoux
Title: Analysis of the exit point distribution using a semi-classical approach
Program information
Abstract
We consider the exit event from a metastable state for a stochastic process. We show how, using the concept of quasi-stationary distribution, one can model the exit event from a metastable state by a jump Markov model. For the overdamped Langevin dynamics and in the limit of small noise, we show that the jump Markov model can be parameterized by the Eyring-Kramers formulas. This mathematical analysis is useful from a modeling and a numerical viewpoint. Indeed, it justifies the use of jump Markov models (kinetic Monte Carlo or Markov State Models) with jump rates determined using the Eyring-Kramers formula (Harmonic Transition State Theory) to describe the evolution of a molecular system over long timescales. This study is also motivated by the design and analysis in terms of accuracy and efficiency of so-called accelerated dynamics algorithms (developed in particular by D. Perez, A.F. Voter and collaborators at Los Alamos National Laboratory), which use the approximation by a jump Markov model to simulate metastable trajectories of the Langevin or overdamped Langevin dynamics over large timescales.
The lectures will be divided into three parts. The first lecture will be devoted to an introduction to the quasi-stationary distribution approach for the exit problem. We will present the main motivation for this approach, the main result on the exit point distribution for the overdamped Langevin, as well as numerical counterparts. The second lecture will provide some essential tools from the semi-classical analysis of Witten Laplacians. We will discuss in particular some results on the spectrum of the infinitesimal generator of the overdamped Langevin dynamics in the small noise limit. Finally, the third lecture will give a detailed description of the first exit point distribution of the overdamped Langevin dynamics in the small noise regime, using in particular the techniques introduced in the second lecture.
References
H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger Operators, Springer, 1986.
M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Cambridge University Press, 1999.
G. Di Gesù, T. Lelièvre, D. Le Peutrec and B. Nectoux, Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach, Faraday Discussion, 195, 469-495, 2016.
G. Di Gesù, T. Lelièvre, D. Le Peutrec and B. Nectoux, Sharp asymptotics of the first exit point density, Annals of PDE, 5(1), 2019.
B. Helffer, M. Klein and F. Nier, Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp. 26, 41-85, 2004.
Zoom address
This is a virtual lecture series. You can access the lectures at ZOOM 937 905 8748 (without password).
Organizer
Insuk Seo (Seoul National University)
Sponsor
Samsung Science & Technology Foundation