The small mass limit is derived for a stochastic N particle system in mean-field limit theory. In the case of the constant communication weight function and by applying the averaging approach to distribution dependent slow–fast stochastic differential equations (which has an independent interest), the small mass limit model is derived.
REFERENCES
1.
D. A.
Dawson
, “Critical dynamics and fluctuations for a mean-field model of cooperative behavior
,” J. Stat. Phys.
31
(1
), 29
–85
(1983
).2.
F.
Golse
, “On the dynamics of large particle systems in the mean field limit
,” in Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity
, Lecture Notes in Applied Mathematics and Mechanics Vol. 3 (Springer
, Cham
, 2016
), pp. 1
–144
.3.
N.
Fournier
, M.
Hauray
, and S.
Mischler
, “Propagation of chaos for the 2D viscous vortex model
,” J. Eur. Math. Soc.
16
, 1423
–1466
(2014
).4.
S.
Serfaty
, “Mean field limit for Coulomb-type flows
,” Duke Math. J.
169
(15
), 2887
–2935
(2020
).5.
A.-S.
Sznitman
, “Topics in propagation of chaos
,” in Ecole d’Été de Probabilités de Saint–Flour XIX—1989
, Lecture Notes in Mathematics Vol. 1464 (Springer
, Berlin
, 1991
), pp. 165
–251
.6.
F.
Cucker
and S.
Smale
, “Emergent behavior in flocks
,” IEEE Trans. Autom. Control
52
, 852
–862
(2007
).7.
F.
Cucker
and S.
Smale
, “On the mathematics of emergence
,” Jpn. J. Math.
2
, 197
–227
(2007
).8.
F.
Bolley
, J. A.
Cañizo
, and J. A.
Carrillo
, “Stochastic mean-field limit: Non-lipschitz forces and swarming
,” Math. Models Methods Appl. Sci.
21
(11
), 2179
–2210
(2011
).9.
J. A.
Carrillo
and Y.-P.
Choi
, “Mean-field limits: From particle descriptions to macroscopic equations
,” Arch. Ration. Mech. Anal.
241
(3
), 1529
–1573
(2021
).10.
P.-E.
Jabin
and Z.
Wang
, “Quantitative estimates of propagation of chaos for stochastic systems with W−1,∞ kernels
,” Invent. Math.
214
, 523
–591
(2018
).11.
S.
Mischler
, C.
Mouhot
, and B.
Wennberg
, “A new approach to quantitative propagation of chaos for drift, diffusion and jump processes
,” Probab. Theory Relat. Fields
161
, 1
–59
(2015
).12.
P.
Monmarché
, “Long-time behaviour and propagation of chaos for mean field kinetic particles
,” Stochastic Process. Appl.
127
, 1721
–1737
(2017
).13.
M.
Freidlin
, “Some remarks on the Smoluchowski–Kramers approximation
,” J. Stat. Phys.
117
(3/4
), 617
–634
(2004
).14.
R. C.
Fetecau
and W.
Sun
, “First-order aggregation models and zero inertia limits
,” J. Differ. Equations
259
(11
), 6774
–6802
(2015
).15.
Y. P.
Choi
and O.
Tse
, “Quantified overdamped limit for kinetic Vlasov–Fokker–Planck equations with singular interaction forces
,” J. Differ. Equations.
330
(5
), 150
–207
(2022
).16.
N.
Fournier
and A.
Guillin
, “On the rate of convergence in Wasserstein distance of the empirical measure
,” Probab. Theory Relat. Fields
162
, 707
–738
(2014
).17.
S.
Hottovy
, A.
McDaniel
, G.
Volpe
, and J.
Wehr
, “The Smoluchowski–Kramers limit of stochastic differential equations with arbitrary state-dependent friction
,” Commun. Math. Phys.
336
, 1259
–1283
(2015
).18.
Y.
Lv
and W.
Wang
, “Smoluchowski–Kramers approximation with state dependent damping and highly random oscillation
,” Discrete Cont. Dyn.-B
(published online 2022
)19.
H.
Huang
, “Quantitative estimate of the overdamped limit for the Vlasov–Fokker–Planck systems
,” Part. Differ. Equations Appl. Math.
4
, 100186
(2021
).20.
W.
E
, D.
Liu
, and E.
Vanden-Eijnden
, “Analysis of multiscale methods for stochastic differential equations
,” Commun. Pure Appl. Math.
58
(11
), 1544
–1585
(2005
).21.
W.
Wang
and A. J.
Roberts
, “Average and deviation for slow–fast stochastic partial differential equations
,” J. Differ. Equations
253
, 1265
–1286
(2012
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
2022
Author(s)
You do not currently have access to this content.