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Estimating Small Probabilities for Langevin Dynamics
TL;DR: In this article, a simplification of Girsanov's formula is obtained in which the relationship between the infinitesimal generator of the underlying diffusion and the change of probability measure corresponding to a change in the potential energy is made explicit.
Abstract: The problem of estimating small transition probabilities for overdamped Langevin dynamics is considered. A simplification of Girsanov's formula is obtained in which the relationship between the infinitesimal generator of the underlying diffusion and the change of probability measure corresponding to a change in the potential energy is made explicit. From this formula an asymptotic expression for transition probability densities is derived. Separately the problem of estimating the probability that a small noise Langevin process excapes a potential well is discussed.
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Book•
27 Mar 1998
TL;DR: The LDP for Abstract Empirical Measures and applications-The Finite Dimensional Case and Applications of Empirically Measures LDP are presented.
Abstract: LDP for Finite Dimensional Spaces.- Applications-The Finite Dimensional Case.- General Principles.- Sample Path Large Deviations.- The LDP for Abstract Empirical Measures.- Applications of Empirical Measures LDP.
5,578 citations
"Estimating Small Probabilities for ..." refers background in this paper
...Though the LDP by itself says nothing about probabilities at a fixed β, the Freidlin-Wentzell theory has Date: April 2012....
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...In particular, the asymptotic behavior of probabilities as β → ∞ satisfy a large deviations principle (LDP) [5]....
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...1 and continuity of ∇V it follows that Aǫ is a continuity set [5] with respect to the rate function φ → ...
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TL;DR: Some Mathematical Preliminaries as mentioned in this paper include the Ito Integrals, Ito Formula and the Martingale Representation Theorem, and Stochastic Differential Equations.
Abstract: Some Mathematical Preliminaries.- Ito Integrals.- The Ito Formula and the Martingale Representation Theorem.- Stochastic Differential Equations.- The Filtering Problem.- Diffusions: Basic Properties.- Other Topics in Diffusion Theory.- Applications to Boundary Value Problems.- Application to Optimal Stopping.- Application to Stochastic Control.- Application to Mathematical Finance.
4,705 citations
"Estimating Small Probabilities for ..." refers background in this paper
...Formally, Xt is a time homogeneous Itō process [1] with conservative drift and constant diffusion....
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Book•
01 Jan 1984
TL;DR: In this article, the authors introduce the concept of random perturbations in Dynamical Systems with a Finite Time Interval (FTI) and the Averaging Principle.
Abstract: 1.Random Perturbations.- 2.Small Random Perturbations on a Finite Time Interval.- 3.Action Functional.- 4.Gaussian Perturbations of Dynamical Systems. Neighborhood of an Equilibrium Point.- 5.Perturbations Leading to Markov Processes.- 6.Markov Perturbations on Large Time Intervals.- 7.The Averaging Principle. Fluctuations in Dynamical Systems with Averaging.- 8.Random Perturbations of Hamiltonian Systems.- 9. The Multidimensional Case.- 10.Stability Under Random Perturbations.- 11.Sharpenings and Generalizations.- References.- Index.
4,070 citations
"Estimating Small Probabilities for ..." refers background in this paper
...Small probabilities of the process (1) have been studied in the large β limit in the context of Freidlin-Wentzell theory [4]....
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Book•
21 Feb 1967
TL;DR: In a course of lectures given by Professor Nelson at Princeton during the spring term of 1966, the authors traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations.
Abstract: These notes are based on a course of lectures given by Professor Nelson at Princeton during the spring term of 1966. The subject of Brownian motion has long been of interest in mathematical probability. In these lectures, Professor Nelson traces the history of earlier work in Brownian motion, both the mathematical theory, and the natural phenomenon with its physical interpretations. He continues through recent dynamical theories of Brownian motion, and concludes with a discussion of the relevance of these theories to quantum field theory and quantum statistical mechanics.
1,517 citations
"Estimating Small Probabilities for ..." refers methods in this paper
...The overdamped version is obtained from a scaling limit of the Langevin equation in which a damping constant tends to infinity [2], [3]....
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Book•
25 Aug 2008TL;DR: General Methods and Algorithms for Generating Random Objects and Output Analysis and Variance-Reduction Methods for Stochastic Optimization.
Abstract: General Methods and Algorithms.- Generating Random Objects.- Output Analysis.- Steady-State Simulation.- Variance-Reduction Methods.- Rare-Event Simulation.- Derivative Estimation.- Stochastic Optimization.- Algorithms for Special Models.- Numerical Integration.- Stochastic Di3erential Equations.- Gaussian Processes.- Levy Processes.- Markov Chain Monte Carlo Methods.- Selected Topics and Extended Examples.- What This Book Is About.- What This Book Is About.
1,265 citations
"Estimating Small Probabilities for ..." refers background in this paper
...[16], [17]), in which one chooses another probability measure P̃ for sampling....
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