Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach

We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.Comment: 14 page

Repartition of the quasi-stationary distribution and first exit point density for a double-well potential

Let f : R d → R be a smooth function and (Xt) t≥0 be the stochastic process solution to the overdamped Langevin dynamics dXt = −−f (Xt)dt + √ h dBt. Let Ω ⊂ R d be a smooth bounded domain and assume that f | Ω is a double-well potential with degenerate barriers. In this work, we study in the small temperature regime, i.e. when h → 0 + , the asymptotic repartition of the quasi-stationary distribution of (Xt) t≥0 in Ω within the two wells of f | Ω. We show that this distribution generically concentrates in precisely one well of f | Ω when h → 0 + but can nevertheless concentrate in both wells when f | Ω admits sufficient symmetries. This phenomenon corresponds to the so-called tunneling effect in semiclassical analysis. We also investigate in this setting the asymptotic behaviour when h → 0 + of the first exit point distribution from Ω of (Xt) t≥0 when X0 is distributed according to the quasi-stationary distribution. 1 Setting and results 1.1 Quasi-stationary distribution and purpose of this work Let (X t) t≥0 be the stochastic process solution to the overdamped Langevin dynamics in R d : dX t = −−f (X t)dt + √ h dB t , (1) where f : R d → R is the potential (chosen C ∞ in all this work), h > 0 is the temperature and (B t) t≥0 is a standard d-dimensional Brownian motion. Let Ω be a C ∞ bounded open and connected subset of R d and introduce τ Ω = inf{t ≥ 0 | X t / ∈ Ω} the first exit time from Ω. A quasi-stationary distribution for the process (1) on Ω is a probability measure µ h on Ω such that, when X 0 ∼ µ h , it holds for any time t > 0 and any Borel set A ⊂ Ω, P(X t ∈ A | t < τ Ω) = µ h (A)

Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $\Omega \subset \mathbb R^d$ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $\Omega$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $\Omega$ through each of the saddle points of $f$ on $\partial \Omega$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics

Law of Large Numbers for Bayesian two-layer Neural Network trained with Variational Inference

We provide a rigorous analysis of training by variational inference (VI) of Bayesian neural networks in the two-layer and infinite-width case. We consider a regression problem with a regularized evidence lower bound (ELBO) which is decomposed into the expected log-likelihood of the data and the Kullback-Leibler (KL) divergence between the a priori distribution and the variational posterior. With an appropriate weighting of the KL, we prove a law of large numbers for three different training schemes: (i) the idealized case with exact estimation of a multiple Gaussian integral from the reparametrization trick, (ii) a minibatch scheme using Monte Carlo sampling, commonly known as Bayes by Backprop, and (iii) a new and computationally cheaper algorithm which we introduce as Minimal VI. An important result is that all methods converge to the same mean-field limit. Finally, we illustrate our results numerically and discuss the need for the derivation of a central limit theorem

Analyse spectrale et analyse semi-classique pour la métastabilité en dynamique moléculaire

This thesis is dedicated to the study of the sharp asymptotic behaviour in the low temperature regime of the exit event from a metastable domain $\Omega\subset \mathbb R^d$ (exit point and exit time) for the overdamped Langevin process. In practice, the overdamped Langevin dynamics can be used to describe for example the motion of the atoms of a molecule or the diffusion of interstitial impurities in a crystal. The obtention of sharp asymptotic approximations of the first exit point density in the small temperature regime is the main result of this thesis. These results justify the use of the Eyring-Kramers law to model the exit event. The Eyring-Kramers law is used for example to compute the transition rates between the states of a system in a kinetic Monte-Carlo algorithm in order to sample efficiently the state-to-state dynamics. The cornerstone of our analysis is the quasi stationary distribution associated with the overdamped Langevin dynamics in $\Omega$. The proofs are based on tools from semi-classical analysis. This thesis is divided into three independent chapters. The first chapter (in French) is dedicated to an introduction to the mathematical results. The other two chapters (in English) are devoted to the precise statements and proofs.Dans cette thèse, nous étudions le comportement asymptotique précis à basse température de l’événement de sortie d'un domaine métastable $\Omega\subset \mathbb R^d$ (point de sortie et temps de sortie) pour le processus de Langevin suramorti. En pratique, le processus de Langevin suramorti peut par exemple simuler l'évolution des positions des atomes d'une molécule ou la diffusion d'impuretés interstitielles dans un cristal. Nos résultats principaux concernent le comportement asymptotique précis de la distribution de la loi du point de sortie de $\Omega$. Dans la limite d'une petite température, ces résultats permettent de justifier l'utilisation de la formule d'Eyring-Kramers pour modéliser les événements de sortie de $\Omega$. La loi d'Eyring-Kramers est par exemple utilisée pour calculer les taux de transition entre les états d'un système dans un algorithme de Monte-Carlo cinétique afin de simuler efficacement les différents états visités par le système. L'analyse repose de manière essentielle sur la distribution quasi stationnaire associée au processus de Langevin suramorti dans $\Omega$. Nos preuves utilisent des outils d'analyse semi-classique. La thèse se décompose en trois chapitres indépendants. Le premier chapitre (rédigé en français) est une introduction aux résultats obtenus. Les deux autres chapitres (rédigées en anglais) sont consacrés aux énoncés mathématiques

Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit

International audienceWe prove a sharp asymptotic formula for the mean exit time from an open bounded domain D ⊂ R d for the overdamped Langevin dynamics dX t = −−f (X t)dt + √ 2ε dB t in the limit ε → 0 and in the case when D contains a unique non degenerate minimum of f and ∂ n f > 0 on ∂D. As a direct consequence, one obtains in the limit ε → 0, a sharp asymptotic estimate of the smallest eigenvalue of the operator L ε = −ε∆ + f · associated with Dirichlet boundary conditions on ∂D. The approach does not require f | ∂D to be a Morse function. The proof is based on results from [7,8] and a formula for the mean exit time from D introduced in the potential theoretic approach to metastability [4, 5]

Mean exit time for the overdamped Langevin process: the case with critical points on the boundary

Let (Xt) t≥0 be the overdamped Langevin process on R d , i.e. the solution of the stochastic differential equation dXt = −∇f (Xt) dt + √ h dBt. Let Ω ⊂ R d be a bounded domain. In this work, when X0 = x ∈ Ω, we derive new sharp asymptotic equivalents (with optimal error terms) in the limit h → 0 of the mean exit time from Ω of the process (Xt) t≥0 when the function f : Ω → R has critical points on the boundary of Ω. The proof is based on recent results from [27] and combines techniques from the potential theory and the large deviations theory. The approach also allows us to provide new sharp leveling results on the mean exit time from Ω

Spectral analysis and semi-classical analysis for metastability in molecular dynamics

Dans cette thèse, nous étudions le comportement asymptotique précis à basse température de l’événement de sortie d'un domaine métastable Ω⊂ℝ^d (point de sortie et temps de sortie) pour le processus de Langevin sur amorti. En pratique, le processus de Langevin sur amorti peut par exemple simuler l'évolution des positions des atomes d'une molécule ou la diffusion d'impuretés interstitielles dans un cristal. Nos résultats principaux concernent le comportement asymptotique précis de la distribution de la loi du point de sortie de Ω. Dans la limite d'une petite température, ces résultats permettent de justifier l'utilisation de la formule d'Eyring-Kramers pour modéliser les événements de sortie de Ω. La loi d'Eyring-Kramers est par exemple utilisée pour calculer les taux de transition entre les états d'un système dans un algorithme de Monte-Carlo cinétique afin de simuler efficacement les différents états visités par le système. L'analyse repose de manière essentielle sur la distribution quasi stationnaire associée au processus de Langevin sur amorti dans Ω. Nos preuves utilisent des outils d'analyse semi-classique. La thèse se décompose en trois chapitres indépendants. Le premier chapitre (rédigé en français) est une introduction aux résultats obtenus. Les deux autres chapitres (rédigées en anglais) sont consacrés aux énoncés mathématiquesThis thesis is dedicated to the study of the sharp asymptotic behaviour in the low temperature regime of the exit event from a metastable domain Ω⊂ℝ^d (exit point and exit time) for the overdamped Langevin process. In practice, the overdamped Langevin dynamics can be used to describe for example the motion of the atoms of a molecule or the diffusion of interstitial impurities in a crystal. The obtention of sharp asymptotic approximations of the first exit point density in the small temperature regime is the main result of this thesis. These results justify the use of the Eyring-Kramers law to model the exit event. The Eyring-Kramers law is used for example to compute the transition rates between the states of a system in a kinetic Monte-Carlo algorithm in order to sample efficiently the state-to-state dynamics. The cornerstone of our analysis is the quasi stationary distribution associated with the overdamped Langevin dynamics in Ω. The proofs are based on tools from semi-classical analysis. This thesis is divided into three independent chapters. The first chapter (in French) is dedicated to an introduction to the mathematical results. The other two chapters (in English) are devoted to the precise statements and proof

Analyse spectrale et analyse semi-classique pour l'étude de la métastabilité en dynamique moléculaire

This thesis is dedicated to the study of the sharp asymptotic behaviour in the low temperature regime of the exit event from a metastable domain Ω⊂ℝ^d (exit point and exit time) for the overdamped Langevin process. In practice, the overdamped Langevin dynamics can be used to describe for example the motion of the atoms of a molecule or the diffusion of interstitial impurities in a crystal. The obtention of sharp asymptotic approximations of the first exit point density in the small temperature regime is the main result of this thesis. These results justify the use of the Eyring-Kramers law to model the exit event. The Eyring-Kramers law is used for example to compute the transition rates between the states of a system in a kinetic Monte-Carlo algorithm in order to sample efficiently the state-to-state dynamics. The cornerstone of our analysis is the quasi stationary distribution associated with the overdamped Langevin dynamics in Ω. The proofs are based on tools from semi-classical analysis. This thesis is divided into three independent chapters. The first chapter (in French) is dedicated to an introduction to the mathematical results. The other two chapters (in English) are devoted to the precise statements and proofsDans cette thèse, nous étudions le comportement asymptotique précis à basse température de l’événement de sortie d'un domaine métastable Ω⊂ℝ^d (point de sortie et temps de sortie) pour le processus de Langevin sur amorti. En pratique, le processus de Langevin sur amorti peut par exemple simuler l'évolution des positions des atomes d'une molécule ou la diffusion d'impuretés interstitielles dans un cristal. Nos résultats principaux concernent le comportement asymptotique précis de la distribution de la loi du point de sortie de Ω. Dans la limite d'une petite température, ces résultats permettent de justifier l'utilisation de la formule d'Eyring-Kramers pour modéliser les événements de sortie de Ω. La loi d'Eyring-Kramers est par exemple utilisée pour calculer les taux de transition entre les états d'un système dans un algorithme de Monte-Carlo cinétique afin de simuler efficacement les différents états visités par le système. L'analyse repose de manière essentielle sur la distribution quasi stationnaire associée au processus de Langevin sur amorti dans Ω. Nos preuves utilisent des outils d'analyse semi-classique. La thèse se décompose en trois chapitres indépendants. Le premier chapitre (rédigé en français) est une introduction aux résultats obtenus. Les deux autres chapitres (rédigées en anglais) sont consacrés aux énoncés mathématique

Mean exit time for the overdamped Langevin process: the case with critical points on the boundary

Let $(X_t)_{t\ge 0}$ be the overdamped Langevin process on $\mathbb R^d$, i.e. the solution of the stochastic differential equation $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t.$$Let $\Omega\subset \mathbb R^d$ be a bounded domain. In this work, when $X_0=x\in \Omega$, we derive new sharp asymptotic equivalents (with optimal error terms) in the limit $h\to 0$ of the mean exit time from $\Omega$ of the process $(X_t)_{t\ge 0}$ (which is the solution of (-\frac h2 \Delta+\nabla f\cdot \nabla)w=1 \text{ in } \Omega \text{ and } w=0 \text{ on } \pa \Omega), when the function $f:\overline \Omega\to \mathbb R$ has critical points on $\partial \Omega$. Such a setting is the one considered in many cases in molecular dynamics simulations. This problem has been extensively studied in the literature but such a setting has never been treated. The proof, mainly based on techniques from partial differential equations, uses recent spectral results from~\cite{pcbord} and its starting point is a formula from the potential theory. We also provide new sharp leveling results on the mean exit time from~$\Omega$