# Estimating Small Probabilities for Langevin Dynamics

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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### "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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4,448 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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3,873 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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1,449 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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1,180 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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