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Large Deviations for the Langevin Equation with Strong Damping

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Abstract

We study large deviations in the Langevin dynamics, with damping of order \(\epsilon ^{-1}\) and noise of order 1, as \(\epsilon \downarrow 0\). The damping coefficient is assumed to be state dependent. We proceed first with a change of time and then we use a weak convergence approach to large deviations and its equivalent formulation in terms of the Laplace principle, to determine the good action functional. Some applications of these results to the exit problem from a domain and to the wave front propagation for a suitable class of reaction diffusion equations are considered.

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References

  1. Dupuis, P., Ellis, R.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics. Wiley, New York (1997)

    Book  MATH  Google Scholar 

  2. Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, Z., Freidlin, M.I.: Smoluchowski–Kramers approximation and exit problems. Stoch. Dyn. 5, 569–585 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Freidlin, M.I.: Functional Integration and Partial Differential Equations. Annals of Mathematics Studies, vol. 109. Princeton University Press, Princeton (1985)

    Google Scholar 

  5. Freidlin, M.I.: Limit theorems for large deviations and reaction–diffusion equations. Ann. Probab. 13, 639–675 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  6. Freidlin, M.I.: Coupled reaction–diffusion equations. Ann. Probab. 19, 29–57 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Freidlin, M.I.: Quasi-deterministic approximation, metastability and stochastic resonance. Phys. D 137, 333–352 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Freidlin, M.I.: Some remarks on the Smoluchowski–Kramers approximation. J. Stat. Phys. 117, 617–634 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Freidlin, M.I., Hu, W.: Smoluchowski–Kramers approximation in the case of variable friction. J. Math. Sci. 179, 184–207 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Freidlin, M.I., Koralov, L.: Nonlinear stochastic perturbations of dynamical systems and quasi-linear parabolic PDE’s with a small parameter. Probab. Theory Relat. Fields 147, 273–301 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, 3rd edn. Springer, Heidelberg (2012)

    Book  MATH  Google Scholar 

  12. Hottovy, S., McDaniel, A., Volpe, G., Wehr, J.: The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction. Commun. Math. Phys. 336, 1259–1283 (2015)

  13. Lyv, Y., Roberts, A.J.: Large deviation principle for singularly perturbed stochastic damped wave equations. Stoch. Anal. Appl. 32, 50–60 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

Sandra Cerrai was partially supported by the NSF Grant DMS 1407615. Mark Freidlin was partially supported by the NSF Grant DMS 1411866.

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  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Sandra Cerrai & Mark Freidlin

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  1. Sandra Cerrai
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  2. Mark Freidlin
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Correspondence to Sandra Cerrai.

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Cerrai, S., Freidlin, M. Large Deviations for the Langevin Equation with Strong Damping. J Stat Phys 161, 859–875 (2015). https://doi.org/10.1007/s10955-015-1346-2

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  • DOI: https://doi.org/10.1007/s10955-015-1346-2

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