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Simultaneous Small Noise Limit for Singularly Perturbed Slow-Fast Coupled Diffusions

  • Siva R. Athreya
  • Vivek S. Borkar
  • K. Suresh Kumar
  • Rajesh SundaresanEmail author
Article
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Abstract

We consider a simultaneous small noise limit for a singularly perturbed coupled diffusion described by
$$\begin{aligned} dX^{\varepsilon }_t= & {} b(X^{\varepsilon }_t, Y^{\varepsilon }_t)dt + \varepsilon ^{\alpha }dB_t, \\ dY^{\varepsilon }_t= & {} - \frac{1}{\varepsilon } \nabla _yU(X^{\varepsilon }_t, Y^{\varepsilon }_t)dt + \frac{s(\varepsilon )}{\sqrt{\varepsilon }} dW_t, \end{aligned}$$
where \(B_t, W_t\) are independent Brownian motions on \({\mathbb R}^d\) and \({\mathbb R}^m\) respectively, \(b : \mathbb {R}^d \times \mathbb {R}^m \rightarrow \mathbb {R}^d\), \(U : \mathbb {R}^d \times \mathbb {R}^m \rightarrow \mathbb {R}\) and \(s :(0,\infty ) \rightarrow (0,\infty )\). We impose regularity assumptions on b, U and let \(0< \alpha < 1.\) When \(s(\varepsilon )\) goes to zero slower than a prescribed rate as \(\varepsilon \rightarrow 0\), we characterize all weak limit points of \(X^{\varepsilon }\), as \(\varepsilon \rightarrow 0\), as solutions to a differential equation driven by a measurable vector field. Under an additional assumption on the behaviour of \(U(x, \cdot )\) at its global minima we characterize all limit points as Filippov solutions to the differential equation.

Keywords

Averaging principle Slow–fast motion Carathéodory solution Filippov solution Small noise limit Nonlinear filter Spectral gap Reversible diffusion 

Mathematics Subject Classification

60J60 60G35 

Notes

Acknowledgements

Research of S.R.A. was supported in part by ISF-UGC grant, research of V.S.B. was supported in part by a J. C. Bose Fellowship, research of K.S.K. was supported in part by the grant MTR/2017/000416 from SERB and research of R.S. was supported in part by RBCCPS-IISc. S.R.A., V.S.B. and R.S. would like to thank the International Centre for Theoretical Sciences (ICTS) for hospitality during the Large deviation theory in statistical physics: Recent advances and future challenges (Code:ICTS/Prog-ldt/2017/8). The authors thank Sanjoy Mitter for pointing out the reference [29], Laurent Miclo and Patrick Cattiaux for suggestions on the spectral gap estimate in Proposition 2.1(b), and Konstantinos Spiliopoulos for pointing out several references in the literature.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Arapostathis, A., Borkar, V.S., Ghosh, M.K.: Ergodic Control of Diffusion Processes, Volume 143 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2012)Google Scholar
  2. 2.
    Athreya, K .B., Hwang, Chii-Ruey: Gibbs measures asymptotics. Sankhya A 72(1), 191–207 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Allinger, D.F., Mitter, Sanjoy K.: New results on the innovations problem for non-linear filtering. Stochastics 4(4), 339–348 (1981)zbMATHCrossRefGoogle Scholar
  4. 4.
    Athreya, S.R., Sunder, V.S.: Measure & Probability. Universities Press, CRC Press, Hyderabad (2008)zbMATHGoogle Scholar
  5. 5.
    Biswas, A., Borkar, Vivek S.: Small noise asymptotics for invariant densities for a class of diffusions: a control theoretic view. J. Math. Anal. Appl. 360(2), 476–484 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bain, A., Crisan, Dan: Fundamentals of stochastic filtering, volume 60 of Stochastic Modelling and Applied Probability, vol. 60. Springer, New York (2009)zbMATHCrossRefGoogle Scholar
  7. 7.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, Markus: Metastability and small eigenvalues in Markov chains. J. Phys. A 33(46), L447–L451 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, M.: Metastability in stochastic dynamics of disordered mean-field models. Prob. Theory Relat. Fields 119(1), 99–161 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bovier, A., Eckhoff, M., Gayrard, V., Klein, Markus: Metastability in reversible diffusion processes. I. Sharp asymptotics for capacities and exit times. J. Eur. Math. Soc. (JEMS) 6(4), 399–424 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bakry, D., Gentil, I., Ledoux, Michel: Analysis and Geometry of Markov Diffusion Operators, Volume 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348. Springer, Cham (2014)Google Scholar
  11. 11.
    Billingsley, Patrick: Convergence of Probability Measures. Wiley, New York (1968)zbMATHGoogle Scholar
  12. 12.
    Buckdahn, R., Ouknine, Y., Quincampoix, M.: On limiting values of stochastic differential equations with small noise intensity tending to zero. Bull. Sci. Math. 133(3), 229–237 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Borkar, Vivek S: Probability Theory. Universitext. Springer, New York (1995). An advanced courseCrossRefGoogle Scholar
  14. 14.
    Chiang, T.-S., Hwang, C.-R., Sheu, Shuenn Jyi: Diffusion for global optimization in \({\bf R}^n\). SIAM J. Control Optim. 25(3), 737–753 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Delarue, F., Flandoli, Franco: The transition point in the zero noise limit for a 1D Peano example. Discret. Contin. Dyn. Syst. Ser. A 34, 4071–4084 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Eckhoff, Michael: Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime. Ann. Prob. 33(1), 244–299 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Eckmann, J.-P., Ruelle, D.: Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3, part 1), 617–656 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Fleming, W.H., Soner, H Mete: Controlled Markov Processes and Viscosity Solutions, Volume 25 of Stochastic Modelling and Applied Probability, second edn. Springer, New York (2006)Google Scholar
  19. 19.
    Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems, volume 260 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, 3rd edition, (2012). Translated from the 1979 Russian original by Joseph SzücsGoogle Scholar
  20. 20.
    Foster, D., Young, Peyton: Stochastic evolutionary game dynamics. Theor. Popul. Biol. 38(2), 219–232 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Gelfand, S.B., Mitter, Sanjoy K.: Recursive stochastic algorithms for global optimization in \({ R}^d\). SIAM J. Control Optim. 29(5), 999–1018 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Gailus, S., Spiliopoulos, Konstantinos: Statistical inference for perturbed multiscale dynamical systems. Stoch. Process. Appl. 127(2), 419–448 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Haussmann, U.G., de Zakai, L’équation: et le problème séparé du contrôle optimal stochastique. In: Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math., pp. 37–62. Springer, Berlin, (1985)Google Scholar
  24. 24.
    Hwang, C.-R., Sheu, Shuenn Jyi: Large-time behavior of perturbed diffusion Markov processes with applications to the second eigenvalue problem for Fokker-Planck operators and simulated annealing. Acta Appl. Math. 19(3), 253–295 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hwang, Chii-Ruey: Laplace’s method revisited: weak convergence of probability measures. Ann. Prob. 8(6), 1177–1182 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, Volume 24 of North-Holland Mathematical Library, 2nd edn. North-Holland Publishing Co., Kodansha, Ltd., Amsterdam (1989)Google Scholar
  27. 27.
    Jongen, H.T., Weber, G.-W.: On parametric non-linear programming. Ann. Opr. Res 27, 253–284 (1990)zbMATHCrossRefGoogle Scholar
  28. 28.
    John, S.S., Biles, Daniel C: A comparison of the Carathéodory and Filippov solution sets. J. Math. Anal. Appl. 198(2), 571–580 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Kurtz, T.G., Ocone, D.L.: Unique characterization of conditional distributions in nonlinear filtering. Ann. Prob. 16(1), 80–107 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kabanov, Y., Pergamenshchikov, S.: Two-Scale Stochastic Systems, Volume 49 of Applications of Mathematics. Asymptotic Analysis and Control, Stochastic Modelling and Applied Probability, vol. 49. Springer, Berlin (2003)zbMATHCrossRefGoogle Scholar
  31. 31.
    Kunita, H.: Cauchy problem for stochastic partial differential equations arising in nonlinear filtering theory. Systems Control Lett. 1(1), 37–41 (1981/82)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Liptser, R.: Large deviations for two scaled diffusions. Prob. Theory Relat. Fields 106(1), 71–104 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Metafune, Giorgio, Pallara, Diego, Rhandi, Abdelaziz: Global properties of invariant measures. J. Funct. Anal. 223(2), 396–424 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Morse, M.R., Spiliopoulos, Konstantinos: Moderate deviations for systems of slow-fast diffusions. Asymptot. Anal. 105,3–4, 97–135 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Puhalskii, A.: On large deviations of coupled diffusions with time scale separation. Ann. Prob. 44(64), 3111–3186 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Priola, E., Wang, F.-Y.: Gradient estimates for diffusion semigroups with singular coefficients. J. Funct. Anal. 236, 244–264 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, vol. 13. Springer, New York (2006)zbMATHGoogle Scholar
  38. 38.
    Sastry, S Shankar: The effects of small noise on implicitly defined nonlinear dynamical systems. IEEE Trans. Circ. Syst. 30(9), 651–663 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Spiliopoulos, Konstantinos: Large deviations and importance sampling for systems of slow-fast motion. Appl. Math. Optim. 67(1), 123–161 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Spiliopoulos, Konstantinos: Fluctuation analysis and short time asymptotics for multiple scales diffusion processes. Stochastics and Dynamics 14(03), 1350026 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Ventcel, A.D.: The asymptotic behavior of the eigenvalues of matrices with elements of the order \({\rm exp}\{-V_{ij}/(2\varepsilon ^{2})\}\). Dokl. Akad. Nauk SSSR 202, 263–265 (1972)MathSciNetGoogle Scholar
  42. 42.
    Veretennikov, AYu.: Large deviations in averaging principle for stochastic differential equation systems (noncompact case). Stoch. Stoch. Rep. 48(1–2), 83–96 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Veretennikov, A.Y.: On large deviations in the averaging principle for SDEs with a “full dependence”. Ann. Prob. 27(1), 284–296 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Veretennikov, A.Y.: On large deviations for SDEs with small diffusion and averaging. Stoch. Process. Appl. 89(1), 69–79 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Veretennikov, Alexander: On large deviations in the averaging principle for SDE’s with a “full dependence”, revisited [mr1681106]. Discret. Contin. Dyn. Syst. Ser. B 18(2), 523–549 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Wong, E.: Representation of martingales, quadratic variation and applications. SIAM J. Control 9, 621–633 (1971)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Stat-Math UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of Electrical EngineeringIndian Institute of TechnologyMumbaiIndia
  3. 3.Department of MathematicsIndian Institute of TechnologyMumbaiIndia
  4. 4.Department of Electrical Communication Engineering and Robert Bosch Centre for Cyber-Physical SystemsIndian Institute of ScienceBangaloreIndia

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