Small mass asymptotic for the motion with vanishing friction

Mark Freidlin; Wenqing Hu; Alexander Wentzell

doi:10.1016/j.spa.2012.08.013

Small mass asymptotic for the motion with vanishing friction

Mark Freidlin, Wenqing Hu, Alexander Wentzell

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13 Scopus citations

Abstract

We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.

Original language	English (US)
Pages (from-to)	45-75
Number of pages	31
Journal	Stochastic Processes and their Applications
Volume	123
Issue number	1
DOIs	https://doi.org/10.1016/j.spa.2012.08.013
State	Published - Jan 2013
Externally published	Yes

Bibliographical note

Funding Information:
This work is supported in part by NSF Grants DMS-0803287 and DMS-0854982 .

Keywords

Boundary theory of Markov processes
Diffusion processes
Smoluchowski-Kramers approximation
Weak convergence

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10.1016/j.spa.2012.08.013

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title = "Small mass asymptotic for the motion with vanishing friction",

abstract = "We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.",

keywords = "Boundary theory of Markov processes, Diffusion processes, Smoluchowski-Kramers approximation, Weak convergence",

author = "Mark Freidlin and Wenqing Hu and Alexander Wentzell",

note = "Funding Information: This work is supported in part by NSF Grants DMS-0803287 and DMS-0854982 .",

year = "2013",

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doi = "10.1016/j.spa.2012.08.013",

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T1 - Small mass asymptotic for the motion with vanishing friction

AU - Freidlin, Mark

AU - Hu, Wenqing

AU - Wentzell, Alexander

N1 - Funding Information: This work is supported in part by NSF Grants DMS-0803287 and DMS-0854982 .

PY - 2013/1

Y1 - 2013/1

N2 - We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.

AB - We consider the small mass asymptotic (Smoluchowski-Kramers approximation) for the Langevin equation with a variable friction coefficient. The friction coefficient is assumed to be vanishing within certain region. We introduce a regularization for this problem and study the limiting motion for the 1-dimensional case and a multidimensional model problem. The limiting motion is a Markov process on a projected space. We specify the generator and the boundary condition of this limiting Markov process and prove the convergence.

KW - Boundary theory of Markov processes

KW - Diffusion processes

KW - Smoluchowski-Kramers approximation

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=84866490125&partnerID=8YFLogxK

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DO - 10.1016/j.spa.2012.08.013

M3 - Article

AN - SCOPUS:84866490125

SN - 0304-4149

VL - 123

SP - 45

EP - 75

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 1

ER -

Small mass asymptotic for the motion with vanishing friction

Abstract

Bibliographical note

Keywords

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