Abstract
We consider a special class of mean field SDEs with common noise which depends on the image of the solution (i.e., the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games; the Feynman–Kac formula is established to solve Schrördinegr type PDEs on \(\mathscr {P}_2\), and the ergodicity is proved for a class of measure-valued diffusion processes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Albeverio,Y. G. Kondratiev, M. Röckner, Differential geometry of Poisson spaces, C. R. Acad. Sci. Paris Sér. I Math. 323(1996), 1129–1134.
L. Ambrosio, G. Savaré, Gradient flows of probability measures, in Handbook of Differential Equations: Evolutionary Equations ( C.M. Dafermos and E. Feireisl Eds.), vol. 3, pp. 1–136, Elsevier, 2007.
P. Cardaliaguet, Notes on mean field games, P.-L. Lions lectures at College de France. Online at https://www.ceremade.dauphine.fr/~cardaliaguet/MFG20130420.pdf.
R. Carmona, F. Delarue, Probabilistic Theory of Mean Field Games with Applications II. Mean-field Games with Common Noise and Master Equations, Springer, 2018.
R. Carmona, F. Delarue, D. Lacker, Mean field games with common noise, Ann. Probab. 44(2016), 3740–3803.
P. Cardaliaguet, F. Delarue, J.-M. Lasry, P.-L. Lions, The Master Equation and the Convergence Problem in Mean Field Games, Princeton University Press, 2019.
F. Delarue, D. Lacker, K. Ramanan, From the master equation to mean field game limit theory: A central limit theorem, Electr. J. of Probab. 24(2019), 1–54.
W. Hammersley, D. Šiška, L. Szpruch, McKean-Vlasov SDE under measure dependent Lyapunov conditions,arXiv:1802.03974v1.
W. R. P. Hammersley, D. \(\breve{S}\)i\(\breve{s}\)ka, L. Szpruch, Weak existence and uniqueness for McKean-Vlasov SDEs with common noise,arXiv:1908.00955
M. Huang, R. Malhamé, P. Caines, Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information & Systems, 2006.
M. Huang, R. Malhamé, P. Caines, Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and de-centralized Nash equilibria, IEEE transactions on automatic control, 2007.
V. Konarovskyi and M.-K. von Renesse, Modied massive arratia flow and Wasserstein diffusion, Comm. Pure Appl. Math. 72(2019).
Y. Kondratiev, E. Lytvynov, A. Vershik, Laplace operators on the cone of Radon measures, J. Funct. Anal. 269(2015), 2947–2976.
V. Konarovskyi, A system of coalescing heavy diffusion particles on the real line, Ann. Probab. 45(2017), 3293–3335.
J.-M. Lasry, P.-L. Lions, Jeux á champ moyen (I): Le cas stationnaire, Comptes Rendus Math. Acad. des Sci. Ser. I 343(2006) 619–625.
J.-M. Lasry, P.-L. Lions, Jeux á champ moyen (II): Horizon finiet contrôle optimal, Comptes Rendus Math. Acad. des Sci. Ser. I 343(2006), 679–684.
J.-M. Lasry, P.-L. Lions, Mean field games, Japan. J. Math. 2(2007), 229–260.
V. Marx, A new approach for the construction of a Wasserstein diffusion, Electr. J. Probab. 23(2018), 1–54.
V. Marx, Innite-dimensional regularization of McKean-Vlasov equation with a Wasserstein diffusion,arXiv:2002.10157.
P. Ren, F.-Y. Wang, Spectral gap for measure-valued diffusion processes, J. Math. Anal. Appl. 483(2020), 123624.
M.-K. von Renesse, K.-T. Sturm, Entropic measure and Wasserstein diffusion, Ann. Probab. 37(2009), 1114–1191.
J. Shao, A new probability measure-valued stochastic process with Ferguson-Dirichlet process as reversible measure, Electr. J. Probab. 16(2011), 271–292.
F.-Y. Wang, Distribution dependent SDEs for Landau type equations, Stoch. Proc. Appl. 128(2018), 595–621.
F.-Y. Wang, Functional inequalities for weighted Gamma distribution on the space of finite measures, Elect. J. Probab. 25(2020), 1–27.
F.-Y. Wang, W. Zhang, Nash inequality for diffusion processes associated with Dirichlet distributions, Front. Math. China 14(2019), 1317–1338.
Acknowledgements
The author is grateful to the referee for valuable suggestions and to Professor Renming Song for helpful conversations.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported in part by NNSFC (11771326, 11831014) and the DFG through the CRC 1283.
Rights and permissions
About this article
Cite this article
Wang, FY. Image-dependent conditional McKean–Vlasov SDEs for measure-valued diffusion processes. J. Evol. Equ. 21, 2009–2045 (2021). https://doi.org/10.1007/s00028-020-00665-z
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-020-00665-z