Rate of convergence in the Smoluchowski-Kramers approximation for mean-field stochastic differential equations

Ta Cong Son; Dung Quang Le; Manh Hong Duong

doi:10.1007/s11118-023-10078-5

Rate of convergence in the Smoluchowski-Kramers approximation for mean-field stochastic differential equations

Ta Cong Son, Dung Quang Le, Manh Hong Duong^*

^*Corresponding author for this work

Mathematics

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Abstract

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the L^p-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

Original language	English
Journal	Potential Analysis
Early online date	3 Jul 2023
DOIs	https://doi.org/10.1007/s11118-023-10078-5
Publication status	E-pub ahead of print - 3 Jul 2023

Keywords

Smoluchowski-Kramers approximation
Stochastic differential by mean-field
Total variation distance
Malliavin calculus

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10.1007/s11118-023-10078-5Licence: Creative Commons: Attribution (CC BY)

SonT2023RateFinal published version, 486 KBLicence: Creative Commons: Attribution (CC BY)

Rigorous coarse-graining of defects at positive temperature
Duong, H. (Principal Investigator)
Engineering & Physical Science Research Council
1/06/22 → 31/05/23
Project: Research Councils
Variational structures, convergence to equilibrium and multiscale analysis for non-Markovian systems
Duong, H. (Principal Investigator)
Engineering & Physical Science Research Council
1/02/22 → 30/06/24
Project: Research Councils

Cite this

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title = "Rate of convergence in the Smoluchowski-Kramers approximation for mean-field stochastic differential equations",

abstract = "In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.",

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author = "Son, {Ta Cong} and Le, {Dung Quang} and Duong, {Manh Hong}",

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doi = "10.1007/s11118-023-10078-5",

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AU - Son, Ta Cong

AU - Le, Dung Quang

AU - Duong, Manh Hong

PY - 2023/7/3

Y1 - 2023/7/3

N2 - In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

AB - In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

KW - Smoluchowski-Kramers approximation

KW - Stochastic differential by mean-field

KW - Total variation distance

KW - Malliavin calculus

U2 - 10.1007/s11118-023-10078-5

DO - 10.1007/s11118-023-10078-5

M3 - Article

SN - 0926-2601

JO - Potential Analysis

JF - Potential Analysis

ER -

Rate of convergence in the Smoluchowski-Kramers approximation for mean-field stochastic differential equations

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Keywords

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Rigorous coarse-graining of defects at positive temperature

Variational structures, convergence to equilibrium and multiscale analysis for non-Markovian systems

Cite this