Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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
Published electronically: March 16, 2016
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Abstract: We prove that on the sphere $ S^{2}$, one can find a sequence of divergence free vector fields $ {\rm b}_{n}$ with the property that the spectral gap of the operators $ A_{{\rm b}_{n}}= \Delta +{\rm 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 $ {\rm b}_{n}\cdot \nabla $. Questions of this type arise when trying to accelerate Markov Monte Carlo methods by adding convergence enhancing motion.

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Additional Information

Brice Franke
Affiliation: Département de Mathématique, UFR Sciences et Techniques, Université de Bretagne Occidentale, 29200 Brest, France
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

DOI: https://doi.org/10.1090/qam/1426
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

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