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A stochastic theory of adiabatic invariance

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Abstract

LetI be a set of invariants for a system of differential equations with an ordero(ε) vector field. When order ε perturbations of zero mean are added to the system we show that, under suitable regularity and ergodicity conditions,I becomes an adiabatic invariant with maximal variations of order one on time scales of order 1/ε2. In the stochastically perturbed case,I behaves asymptotically (for small ε) like a diffusion process on 1/ε2 time scales. The results also apply to an interesting class of deterministic perturbations. This study extends the results of Khas'minskii on stochastically averaged systems, as well as some of the deterministic methods of averaging, to such invariants.

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References

  1. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  2. Borodin, A.N.: A limit theorem for solutions of differential equations with random right-hand side. Theory Prob. Appl.22, 482–497 (1977)

    Article  Google Scholar 

  3. Cogburn, R., Ellison, J.A.: Particle motion in a rapidly varying field. 1990 (submitted)

  4. Cogburn, R., Ellison, J.A., Newberger, B.S., Shih, H.J.: private communication

  5. Dôme, G.: Diffusion due to RF noise. CERN Advanced Accelerator School, Advanced Accelerator Physics, CERN Report No 87-03 (1987), pp. 370–401

    Google Scholar 

  6. Doob, J.L.: Stochastic processes, New York: Wiley 1956

    Google Scholar 

  7. Dumas, H.S., Ellison, J.A., Sáenz, A.W.: Axial channeling in perfect crystals, the continuum model and the method of averaging. Ann. Phys.209, 97–123 (1991)

    Article  Google Scholar 

  8. Ellison, J.A., Sáenz, A.W., Dumas, H.S.: ImprovedN th order averaging theory for periodic systems. J. Diff. Eq.84, 383 (1990)

    Article  Google Scholar 

  9. Fink, A.M.: Almost periodic differential equations. Lecture Notes in Mathematics vol.377, Berlin, Heidelberg, New York: Springer 1974

    Google Scholar 

  10. Freidlin, M.I., Wentzell, A.D.: Random perturbations of dynamical systems. Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  11. Hale, J.K.: Ordinary differential equations. New York: Wiley 1969

    Google Scholar 

  12. Hardy, G.H., Wright, E.M.: An introduction to the theory of numbers. Oxford: Oxford University Press, fifth edition, 1979

    Google Scholar 

  13. Khas'minskii, R.Z.: On processes defined by differential equations with a small parameter. Theory Prob. Appl.11, 211–228 (1966)

    Article  Google Scholar 

  14. Khas'minskii, R.Z.: A limit theorem for solutions of differential equations with random right-hand side. Theory Prob. Appl.11, 390–406 (1966)

    Article  Google Scholar 

  15. Knapp, R., Papanicolaou, G., White, B.: Nonlinearity and localization in one dimensional random media. In: Disorder and Nonlinearity. Bishop, A.R., Campbell, D.K., Pnevmaticas, S. (eds.). Proceedings in Physics vol.39, pp. 2–26. Berlin, Heidelberg, New York: Springer 1989 See also Knapp, R.J.: Nonlinearity and localization in one dimensional random media. Ph.D. thesis, New York University, 1988

    Google Scholar 

  16. Skorokhod, A.V.: Studies in the theory of random processes. Reading, MA: Addison-Wesley 1965

    Google Scholar 

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Communicated by J.L. Lebowitz

Supported by NSF grant DMR-8704348

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Cogburn, R., Ellison, J.A. A stochastic theory of adiabatic invariance. Commun.Math. Phys. 149, 97–126 (1992). https://doi.org/10.1007/BF02096625

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  • DOI: https://doi.org/10.1007/BF02096625

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