Hostname: page-component-77f85d65b8-7lfxl Total loading time: 0 Render date: 2026-03-26T20:44:40.620Z Has data issue: false hasContentIssue false
  • English
  • Français

STABILITY ANALYSIS FOR STOCHASTIC MCKEAN–VLASOV EQUATION

Part of: Special Collection in honour of Professor Anthony Roberts Probability theory on algebraic and topological structures Distribution theory - Probability

Published online by Cambridge University Press:  06 November 2024

C. SHI Open the ORCID record for C. SHI [Opens in a new window]  and
W. WANG Open the ORCID record for W. WANG [Opens in a new window]
C. SHI
Affiliation:
Department of Mathematics, Nanjing University of Science and Technology, Nanjing, China; e-mail: njueduscg@163.com
W. WANG*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, China
*
e-mail: wangweinju@nju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

The pth ($p\geq 1$) moment exponential stability, almost surely exponential stability and stability in distribution for stochastic McKean–Vlasov equation are derived based on some distribution-dependent Lyapunov function techniques.

Keywords

McKean–Vlasov equationpth moment exponential stabilityalmost surely exponential stabilitystability in distribution

MSC classification

Primary: 60B10: Convergence of probability measures
Secondary: 60E05: Distributions

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e6
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bahlali, K., Mezerdi, M. A. and Mezerdi, B., “Stability of McKean–Vlasov stochastic differential equations and applications”, Stoch. Dyn. 20 (2020) Article ID: 2050007; doi:10.1142/S0219493720500070.CrossRefGoogle Scholar
Bao, J. H., Hou, Z. T. and Yuan, C. G., “Stability in distribution of neutral stochastic differential delay equations with Markovian switching”, Stat. Probab. Lett. 79 (2009) 1663–1673; doi:10.1016/j.spl.2009.04.006.CrossRefGoogle Scholar
Brezis, H., Functional analysis, Sobolev spaces and partial differential equations (Springer, New York, 2010).Google Scholar
Cardaliaguet, P., Notes on mean field games (from P. L. Lion’s lectures at College de France). https://www.ceremade.dauphine.fr/cardalia/MFG100629.pdf.Google Scholar
Ding, X. J. and Qiao, H. J., “Stability for stochastic McKean–Vlasov equations with non-Lipschitz coefficients”, SIAM J. Control Optim. 59 (2021) Article ID: 19M1289418; doi:10.1137/19M1289418.CrossRefGoogle Scholar
Du, N. H., Dang, N. H. and Dieu, N. T., “On stability in distribution of stochastic differential delay equations with Markovian switching”, Syst. Control Lett. 65 (2014) 43–49; doi:10.1016/j.sysconle.2013.12.006.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G., Markov processes: characterization and convergence (John Wiley & Sons, NJ, 1986).CrossRefGoogle Scholar
Fei, C., Fei, W. Y., Deng, S. N. and Mao, X. R., “Asymptotic stability in distribution of highly nonlinear stochastic differential equations with G-Brownian motion”, Qual. Theory Dyn. Syst. 22 (2023) Article ID: 57; doi:10.1007/s12346-023-00760-9.CrossRefGoogle Scholar
Hammersley, W. R. P., Siska, D. and Szpruch, L., “McKean–Vlasov SDEs under measure dependent Lyapunov conditions”, Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021) 1032–1057; doi:10.1214/20-AIHP1106.CrossRefGoogle Scholar
Hong, W., Li, S. H. and Liu, W., “Large deviation principle for McKean–Vlasov quasilinear stochastic evolution equations”, Appl. Math. Optim. 84 (2021) 1119–1147; doi:10.1007/s00245-021-09796-2.CrossRefGoogle Scholar
Kallenberg, O., Foundations of modern probability (Springer, New York, 2000).Google Scholar
Khasminskii, R., Stochastic stability of differential equations (Springer, Berlin–Heidelberg, 2011).Google Scholar
Liu, Z. X. and Ma, J., “Existence, uniqueness and ergodicity for McKean–Vlasov SDEs under distribution-dependent Lyapunov conditions”, Preprint, 2023, arXiv:2309.05411.CrossRefGoogle Scholar
Lv, G. Y. and Shan, Y. Q., “Long time behavior of stochastic McKean–Vlasov equations”, Appl. Math. Lett. 128 (2022) Article ID: 107879; doi:10.1016/j.aml.2021.107879.CrossRefGoogle Scholar
McKean, H. P., “A class of Markov processes associated with nonlinear parabolic equations”, Proc. Natl. Acad. Sci. USA 56 (1966) 1907–1911; https://www.jstor.org/stable/57643.CrossRefGoogle ScholarPubMed
Peng, S. G., “G–Brownian motion and dynamic risk measure under volatility uncertainty”, Preprint, 2007, arXiv:0711.2834.Google Scholar
Reis, D., Salkeld, G. and Tugaut, W., “Freidlin–Wentzell LDPs in path space for McKean–Vlasov equations and the functional iterated logarithm law”, Ann. Appl. Probab. 29 (2019) 1487–1540; https://www.jstor.org/stable/26729307.Google Scholar
Vlasov, A. A., “The vibrational properties of an electron gas”, Sov. Phys. 10 (1968) 721–733; doi:10.1070/PU1968v010n06ABEH003709.CrossRefGoogle Scholar
Wang, F. Y., “Distribution dependent SDEs for Landau type equations”, Stoch. Process. Appl. 128 (2018) 596–621; doi:10.1016/j.spa.2017.05.006.CrossRefGoogle Scholar
Wu, H., Hu, J. H., Gao, S. B. and Yuan, C. G., “Stabilization of stochastic McKean–Vlasov equations with feedback control based on discrete-time state observation”, SIAM J. Control Optim. 60 (2022) 2884–2901; doi:10.1137/21M1454997.CrossRefGoogle Scholar
Yuan, C. G. and Mao, X. R., “Asymptotic stability in distribution of stochastic differential equations with Markovian switching”, Stoch. Process. Appl. 103 (2003) 277–291; doi:10.1016/S0304-4149(02)00230-2.CrossRefGoogle Scholar
Yuan, C. G., Zou, J. Z. and Mao, X. R., “Stability in distribution of stochastic differential delay equations with Markovian switching”, Syst. Control Lett. 50 (2003) 195–207; doi:10.1016/S0167-6911(03)00154-3.CrossRefGoogle Scholar