Stochastic averaging with a flattened Hamiltonian: A Markov process on a stratified space (a whiskered sphere)

Author:
Richard B. Sowers

Journal:
Trans. Amer. Math. Soc. **354** (2002), 853-900

MSC (1991):
Primary 60F17; Secondary 37J40, 58A35, 60J35

Published electronically:
October 4, 2001

MathSciNet review:
1867364

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider a random perturbation of a 2-dimensional Hamiltonian ODE. Under an appropriate change of time, we identify a reduced model, which in some aspects is similar to a stochastically averaged model. The novelty of our problem is that the set of critical points of the Hamiltonian has an interior. Thus we can stochastically average outside this set of critical points, but inside we can make no model reduction. The result is a Markov process on a stratified space which looks like a whiskered sphere (i.e, a 2-dimensional sphere with a line attached). At the junction of the sphere and the line, glueing conditions identify the behavior of the Markov process.

**1.**Milton Abramowitz and Irene A. Stegun,*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR**0167642****2.**Richard F. Bass and Krzysztof Burdzy,*Fiber Brownian motion and the “hot spots” problem*, Duke Math. J.**105**(2000), no. 1, 25–58. MR**1788041**, 10.1215/S0012-7094-00-10512-1**3.**A. N. Borodin,*A limit theorem for the solutions of differential equations with a random right-hand side*, Teor. Verojatnost. i Primenen.**22**(1977), no. 3, 498–512 (Russian, with English summary). MR**0517995****4.**A. N. Borodin and M. I. Freidlin,*Fast oscillating random perturbations of dynamical systems with conservation laws*, Ann. Inst. H. Poincaré Probab. Statist.**31**(1995), no. 3, 485–525 (English, with English and French summaries). MR**1338450****5.**Stewart N. Ethier and Thomas G. Kurtz,*Markov processes*, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR**838085****6.**Lawrence C. Evans,*Partial differential equations*, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR**1625845****7.**S. N. Evans and R. B. Sowers,*Pinching and twisting Markov processes*, submitted.**8.**Mark Freidlin and Matthias Weber,*Random perturbations of nonlinear oscillators*, Ann. Probab.**26**(1998), no. 3, 925–967. MR**1634409**, 10.1214/aop/1022855739**9.**Mark Freidlin and Matthias Weber,*A remark on random perturbations of the nonlinear pendulum*, Ann. Appl. Probab.**9**(1999), no. 3, 611–628. MR**1722275**, 10.1214/aoap/1029962806**10.**Mark I. Freidlin and Alexander D. Wentzell,*Random perturbations of Hamiltonian systems*, Mem. Amer. Math. Soc.**109**(1994), no. 523, viii+82. MR**1201269**, 10.1090/memo/0523**11.**I. I. Gihman,*Concerning a theorem of N. N. Bogolyubov*, Ukrain. Mat. Ž.**4**(1952), 215–219 (Russian). MR**0075386****12.**Mark Goresky and Robert MacPherson,*Stratified Morse theory*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR**932724****13.**R. Z. Has′minskiĭ,*Diffusion processes with a small parameter*, Izv. Akad. Nauk SSSR Ser. Mat.**27**(1963), 1281–1300 (Russian). MR**0169278****14.**R. Z. Has′minskiĭ,*A limit theorem for solutions of differential equations with a random right hand part*, Teor. Verojatnost. i Primenen**11**(1966), 444–462 (Russian, with English summary). MR**0203789****15.**R. Z. Has′minskiĭ,*Stochastic processes defined by differential equations with a small parameter*, Teor. Verojatnost. i Primenen**11**(1966), 240–259 (Russian, with English summary). MR**0203788****16.**A. I. Neĭshtadt,*Probability phenomena due to separatrix crossing*, Chaos**1**(1991), no. 1, 42–48. MR**1135894**, 10.1063/1.165816**17.**F. W. J. Olver,*Asymptotics and special functions*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR**0435697**

Frank W. J. Olver,*Asymptotics and special functions*, AKP Classics, A K Peters, Ltd., Wellesley, MA, 1997. Reprint of the 1974 original [Academic Press, New York; MR0435697 (55 #8655)]. MR**1429619****18.**G. C. Papanicolaou and W. Kohler,*Asymptotic theory of mixing stochastic ordinary differential equations*, Comm. Pure Appl. Math.**27**(1974), 641–668. MR**0368142****19.**George C. Papanicolaou and Werner Kohler,*Asymptotic analysis of deterministic and stochastic equations with rapidly varying components*, Comm. Math. Phys.**45**(1975), no. 3, 217–232. MR**0413265****20.**G. C. Papanicolaou, D. Stroock, and S. R. S. Varadhan,*Martingale approach to some limit theorems*, Papers from the Duke Turbulence Conference (Duke Univ., Durham, N.C., 1976), Paper No. 6, Duke Univ., Durham, N.C., 1977, pp. ii+120 pp. Duke Univ. Math. Ser., Vol. III. MR**0461684****21.**Daniel Revuz and Marc Yor,*Continuous martingales and Brownian motion*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1991. MR**1083357****22.**Clark Robinson,*Dynamical systems*, second ed., CRC Press, Boca Raton, FL, 1999, Stability, symbolic dynamics, and chaos. CMP**2001:03****23.**R. L. Stratonovich,*Topics in the theory of random noise. Vol. I: General theory of random processes. Nonlinear transformations of signals and noise*, Revised English edition. Translated from the Russian by Richard A. Silverman, Gordon and Breach Science Publishers, New York-London, 1963. MR**0158437****24.**Daniel W. Stroock and S. R. Srinivasa Varadhan,*Multidimensional diffusion processes*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR**532498****25.**G. Wolansky,*Limit theorem for a dynamical system in the presence of resonances and homoclinic orbits*, J. Differential Equations**83**(1990), no. 2, 300–335. MR**1033190**, 10.1016/0022-0396(90)90060-3

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
60F17,
37J40,
58A35,
60J35

Retrieve articles in all journals with MSC (1991): 60F17, 37J40, 58A35, 60J35

Additional Information

**Richard B. Sowers**

Affiliation:
Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801

Email:
r-sowers@math.uiuc.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-01-02903-8

Keywords:
Markov processes,
stochastic averaging,
stratified space

Received by editor(s):
September 7, 2000

Received by editor(s) in revised form:
June 1, 2001

Published electronically:
October 4, 2001

Additional Notes:
This work was supported by NSF DMS 9615877 and NSF DMS 0071484. The author would also like to thank Professor Sri Namachchivaya of the Department of Aeronautical and Astronautical Engineering at the University of Illinois at Urbana-Champaign for the seemingly infinite time he donated to discussing the contents of this paper and without whose interest this subject would not have been considered. The author would also like to thank Professor Eugene Lerman of the Department of Mathematics at the University of Illinois at Urbana-Champaign for several helpful discussions about stratified spaces.

Article copyright:
© Copyright 2001
American Mathematical Society