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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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TL;DR: In this paper, it was shown that the K. Ito process is a Wiener process if it satisfies conditions (1.1) and (2.2) of this paper.
Abstract: Let $X\{ x(t,w),{\bf P}\} $ be a stochastic process in n-dimensi onalEuclidean space $R_n $ having continuous trajectories, which satisfy the stochastic equation: $x^i (t,\omega ) = x^i (0,\omega ) + \int_0^t {\phi _j^i (s,\omega )} + \int_0^t {\Psi ^i (s,\omega )ds} ,\quad 0 \leqq t \leqq 1.$Here $p = p(d\omega )$ is a measure in the space $\Omega $ of elementary events, $\int_0^t {\Phi _j^i d\xi ^j } $ is considered to be the stochastic integral of K. Ito with respect to the Wiener process $\xi $. The process is called a Wiener process if it satisfies conditions (1.1) and (1.2) of this paper. Process X is called an K. Ito process (with respect to the Wiener process $\xi $) corresponding to the diffusion matrix $\phi (t,\omega ) = ||\phi _j^i (t,\omega )||$ and to the translation vector $\Psi (t,\omega ) = \{ \Psi ^i (t,\omega )\} $.It is proved with certain restrictions imposed on the vector $\varphi (t,\omega ) = \{ \varphi ^i (t,\omega )\} $ that the process $\tilde X = \{ x(t,\omega ),\tilde {\bf P}\...
701 citations
"Estimating Small Probabilities for ..." refers background in this paper
...In general this question is answered by Girsanov’s theorem [11], [12]....
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29 Nov 2010
TL;DR: This book presents a unified theory of rare event simulation and the variance reduction technique known as importance sampling from the point of view of the probabilistic theory of large deviations.
Abstract: This book presents a unified theory of rare event simulation and the variance reduction technique known as importance sampling from the point of view of the probabilistic theory of large deviations. It allows us to view a vast assortment of simulation problems from a unified single perspective.
519 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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Book•
10 Jun 2010
TL;DR: Sampling Methods Stochastic Differential Equations Meta-Stability Free Energy Perturbation Thermodynamic Integration Constrained Dynamics Non-Equilibrium Methods Fluctuation Identities Jarzynski Identity Adaptive Techniques Long Time Convergence Replica Selection Methods Selection Mechanisms Parallel Computation.
Abstract: Sampling Methods Stochastic Differential Equations Meta-Stability Free Energy Perturbation Thermodynamic Integration Constrained Dynamics Non-Equilibrium Methods Fluctuation Identities Jarzynski Identity Adaptive Techniques Long Time Convergence Replica Selection Methods Selection Mechanisms Parallel Computation.
412 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•
01 Jan 2002
TL;DR: This chapter discusses Stochastic Processes, a type of probability theory, and Monte Carlo Simulation, which addresses the problem of uncertainty in deterministic systems.
Abstract: Introduction * Probability Theory * Stochastic Processes * Ito's Formula and Stochastic Differential Equations * Monte Carlo Simulation * Deterministic System and Input * Deterministic System and Stochastic Input * Stochastic System and Deterministic Input * Stochastic System and Stochastic Input * Bibliography * Index
357 citations
"Estimating Small Probabilities for ..." refers background or methods in this paper
...Although Girsanov’s formula and Itō’s lemma can be used with any Itō process [11], in the proof of Theorem 2....
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...pt(x, y) satisfies the PDE ∂ ∂t pt(x, y) = L ∗ V pt(x, y) (4) This is the Fokker-Planck equation [11]....
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...In general this question is answered by Girsanov’s theorem [11], [12]....
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TL;DR: A survey of boundary crossing probabilities and related statistical applications can be found in this paper, where a method for computing first passage distributions of Brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries.
Abstract: : This paper surveys recent results involving boundary crossing probabilities and related statistical applications. The first part is concerned with problems of sequential analysis, especially repeated significance tests and their application to sequential clinical trials involving survival data. The second part develops the probability theory motivated by the problems of Part 1. A method for computing first passage distributions of Brownian motion to linear boundaries is introduced and then modified to handle problems in discrete time and those involving nonlinear boundaries. The third part is concerned with fixed sample statistical problems, especially change-point problems, which involve boundary crossing probabilities. Examples are given of problems for which methods of Part 2 appear adequate and of problems which require new methods.
257 citations
"Estimating Small Probabilities for ..." refers background in this paper
...With X t the kth component of Xt, a well-known formula of Siegmund ([13], [14]) leads to P (...
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