The behavior of the spectral gap under growing drift

Franke, B.; Hwang, C.-R.; Pai, H.-M.; Sheu, S.-J.

doi:10.1090/S0002-9947-09-04939-3

The behavior of the spectral gap under growing drift
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by B. Franke, C.-R. Hwang, H.-M. Pai and S.-J. Sheu

Trans. Amer. Math. Soc. 362 (2010), 1325-1350

DOI: https://doi.org/10.1090/S0002-9947-09-04939-3

Published electronically: October 1, 2009

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Abstract:

We analyze the behavior of the spectral gap of the Laplace- Beltrami operator on a compact Riemannian manifold when a divergence-free drift vector field is added. We increase the drift by multiplication with a large constant $c$ and ask the question how the spectral gap behaves as $c$ goes to infinity. It turns out that the spectral gap stays bounded if and only if the drift-vector field has eigenfunctions in $H^1$. In that case the spectral gaps converge and we determine the limit.

References

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov. 90 (1967), 209 (Russian). MR 0224110
Hassan Aref, The development of chaotic advection, Phys. Fluids 14 (2002), no. 4, 1315–1325. MR 1900232, DOI 10.1063/1.1458932
Henri Berestycki, François Hamel, and Nikolai Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), no. 2, 451–480. MR 2140256, DOI 10.1007/s00220-004-1201-9
R. N. Bhattacharya, V. K. Gupta, and H. F. Walker, Asymptotics of solute dispersion in periodic porous media, SIAM J. Appl. Math. 49 (1989), no. 1, 86–98. MR 978827, DOI 10.1137/0149005
S. Brooks, Markov chain Monte Carlo method and applications, The Statistician, 47 (1998) 69-100.
Mu-Fa Chen, Eigenvalues, inequalities, and ergodic theory, Probability and its Applications (New York), Springer-Verlag London, Ltd., London, 2005. MR 2105651
Mu Fa Chen and Feng Yu Wang, Estimation of the first eigenvalue of second order elliptic operators, J. Funct. Anal. 131 (1995), no. 2, 345–363. MR 1345035, DOI 10.1006/jfan.1995.1092
Mu-Fa Chen and Feng-Yu Wang, Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1239–1267. MR 1401516, DOI 10.1090/S0002-9947-97-01812-6
P. Constantin, A. Kiselev, L. Ryzhik, and A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2) 168 (2008), no. 2, 643–674. MR 2434887, DOI 10.4007/annals.2008.168.643
I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
Albert Fannjiang and George Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 (1994), no. 2, 333–408. MR 1265233, DOI 10.1137/S0036139992236785
Brice Franke, Integral inequalities for the fundamental solutions of diffusions on manifolds with divergence-free drift, Math. Z. 246 (2004), no. 1-2, 373–403. MR 2031461, DOI 10.1007/s00209-003-0604-1
S. Gallot, D. Hulin, and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin, 1987. MR 909697, DOI 10.1007/978-3-642-97026-9
W.K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrica, 57 (1970), 97-109.
Steffen Heinze, Diffusion-advection in cellular flows with large Peclet numbers, Arch. Ration. Mech. Anal. 168 (2003), no. 4, 329–342. MR 1994746, DOI 10.1007/s00205-003-0256-7
Chii-Ruey Hwang, Shu-Yin Hwang-Ma, and Shuenn Jyi Sheu, Accelerating Gaussian diffusions, Ann. Appl. Probab. 3 (1993), no. 3, 897–913. MR 1233633
Chii-Ruey Hwang, Shu-Yin Hwang-Ma, and Shuenn-Jyi Sheu, Accelerating diffusions, Ann. Appl. Probab. 15 (2005), no. 2, 1433–1444. MR 2134109, DOI 10.1214/105051605000000025
C.R. Hwang and H. M. Pai, Blowing up spectral gap of Laplacian on $N$-torus by antisymmetric perturbations, preprint, (2006).
S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, Optimization by simulated annealing, Science 220 (1983), no. 4598, 671–680. MR 702485, DOI 10.1126/science.220.4598.671
Wilhelm Klingenberg, Riemannian geometry, de Gruyter Studies in Mathematics, vol. 1, Walter de Gruyter & Co., Berlin-New York, 1982. MR 666697
H. J. Kushner, Asymptotic global behavior for stochastic approximation and diffusions with slowly decreasing noise effects: global minimization via Monte Carlo, SIAM J. Appl. Math. 47 (1987), no. 1, 169–185. MR 873242, DOI 10.1137/0147010
Peter Li and Shing Tung Yau, Estimates of eigenvalues of a compact Riemannian manifold, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 205–239. MR 573435
David Márquez, Convergence rates for annealing diffusion processes, Ann. Appl. Probab. 7 (1997), no. 4, 1118–1139. MR 1484800, DOI 10.1214/aoap/1043862427
N. Metropolis and S. Ulam, A property of randomness of an arithmetical function, Amer. Math. Monthly 60 (1953), 252–253. MR 53416, DOI 10.2307/2307436
J. M. Ottino, Mixing, chaotic advection, and turbulence, Annual review of fluid mechanics, Vol. 22, Annual Reviews, Palo Alto, CA, 1990, pp. 207–253. MR 1043921
Robert D. Richtmyer, Principles of advanced mathematical physics. Vol. I, Texts and Monographs in Physics, Springer-Verlag, New York-Heidelberg, 1978. MR 517399, DOI 10.1007/978-3-642-46378-5
B. Shraiman and E. Siggia, Scalar turbulence, Nature, 405(2000), 639-646.
Stephen Wiggins and Julio M. Ottino, Foundations of chaotic mixing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 362 (2004), no. 1818, 937–970. MR 2107642, DOI 10.1098/rsta.2003.1356