On how to use drift to push the spectral gap of a diffusion on $S^{2}$ to infinity
Authors:
Brice Franke and Nejib Yaakoubi
Journal:
Quart. Appl. Math. 74 (2016), 321-335
MSC (2010):
Primary 35K05, 60J60, 47A10
DOI:
https://doi.org/10.1090/qam/1426
Published electronically:
March 16, 2016
MathSciNet review:
3505606
Full-text PDF Free Access
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Additional Information
Abstract: We prove that on the sphere $S^{2}$, one can find a sequence of divergence free vector fields $\textrm {b}_{n}$ with the property that the spectral gap of the operators $A_{\textrm {b}_{n}}= \Delta +\textrm {b}_{n}\cdot \nabla$ goes to infinity. The proof uses some suitable adapted Faber-Krahn type inequality for functions which are in the kernel of the operator $\textrm {b}_{n}\cdot \nabla$. Questions of this type arise when trying to accelerate Markov Monte Carlo methods by adding convergence enhancing motion.
References
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- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
- 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
- B. Franke, C.-R. Hwang, H.-M. Pai, and S.-J. Sheu, The behavior of the spectral gap under growing drift, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1325–1350. MR 2563731, DOI 10.1090/S0002-9947-09-04939-3
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- 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
- Hui-Ming Pai and Chii-Ruey Hwang, Accelerating Brownian motion on $N$-torus, Statist. Probab. Lett. 83 (2013), no. 5, 1443–1447. MR 3041295, DOI 10.1016/j.spl.2013.02.009
- Huyi Hu, Yakov Pesin, and Anna Talitskaya, Every compact manifold carries a hyperbolic Bernoulli flow, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 347–358. MR 2093309
- William P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. Sobolev spaces and functions of bounded variation. MR 1014685, DOI 10.1007/978-1-4612-1015-3
References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957 (56 \#9247)
- Pierre H. Bérard, Spectral geometry: direct and inverse problems, with appendixes by Gérard Besson, and by Bérard and Marcel Berger, Lecture Notes in Mathematics, vol. 1207, Springer-Verlag, Berlin, 1986. MR 861271 (88f:58146)
- Isaac Chavel, Eigenvalues in Riemannian geometry, including a chapter by Burton Randol; with an appendix by Jozef Dodziuk, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. MR 768584 (86g:58140)
- 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 (2009e:58045), DOI 10.4007/annals.2008.168.643
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325 (41 \#1976)
- 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 (2005e:58060), DOI 10.1007/s00209-003-0604-1
- B. Franke, C.-R. Hwang, H.-M. Pai, and S.-J. Sheu, The behavior of the spectral gap under growing drift, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1325–1350. MR 2563731 (2010m:35343), DOI 10.1090/S0002-9947-09-04939-3
- Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490 (83e:34002)
- Chii-Ruey Hwang, Shu-Yin Hwang-Ma, and Shuenn-Jyi Sheu, Accelerating diffusions, Ann. Appl. Probab. 15 (2005), no. 2, 1433–1444. MR 2134109 (2006e:60113), DOI 10.1214/105051605000000025
- Hui-Ming Pai and Chii-Ruey Hwang, Accelerating Brownian motion on $N$-torus, Statist. Probab. Lett. 83 (2013), no. 5, 1443–1447. MR 3041295, DOI 10.1016/j.spl.2013.02.009
- Huyi Hu, Yakov Pesin, and Anna Talitskaya, Every compact manifold carries a hyperbolic Bernoulli flow, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 347–358. MR 2093309 (2005g:37062)
- William P. Ziemer, Weakly differentiable functions, Sobolev spaces and functions of bounded variation, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989. MR 1014685 (91e:46046), DOI 10.1007/978-1-4612-1015-3
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Additional Information
Brice Franke
Affiliation:
Département de Mathématique, UFR Sciences et Techniques, Université de Bretagne Occidentale, 29200 Brest, France
MR Author ID:
728183
Email:
Brice.Franke@univ-brest.fr
Nejib Yaakoubi
Affiliation:
Département de Mathématique, Faculté des Sciences de Sfax, Université de Sfax, 3000 Sfax, Tunisia
Email:
nejibyaakoubi@gmail.com
Received by editor(s):
September 12, 2014
Received by editor(s) in revised form:
December 12, 2014
Published electronically:
March 16, 2016
Additional Notes:
The second author visited Brest (France) from November 2013 to January 2014 with a doctoral exchange grant from Université de Bretagne Occidentale. Two more visits (May-June 2013) and (May-June 2014) were made possible through financial support from the École Doctorale de Sfax.
Article copyright:
© Copyright 2016
Brown University