Convergence in Monge-Wasserstein Distance of Mean Field Systems with Locally Lipschitz Coefficients
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Convergence in Monge-Wasserstein Distance of Mean Field Systems with Locally Lipschitz Coefficients Dung Tien Nguyen1
· Son Luu Nguyen2 · Nguyen Huu Du3
Received: 24 July 2018 / Revised: 25 August 2019 / Accepted: 3 October 2019 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract This paper focuses on stochastic systems of weakly interacting particles whose dynamics depend on the empirical measures of the whole populations. The drift and diffusion coefficients of the dynamical systems are assumed to be locally Lipschitz continuous and satisfy global linear growth condition. The limits of such systems as the number of particles tends to infinity are studied, and the rate of convergence of the sequences of empirical measures to their limits in terms of p th Monge-Wasserstein distance is established. We also investigate the existence, uniqueness, and boundedness, and continuity of solutions of the limiting McKean-Vlasov equations associated to the systems. Keywords Mean-field model · Stochastic differential equation · McKean-Vlasov equation · Convergence Mathematics Subject Classification (2010) 60F25 · 60J60 · 93E03
1 Introduction Mean-field models, stochastic systems involving a large number of particles with weak interactions, are first studied in statistical physics, and then vastly studied in many different Dung Tien Nguyen
[email protected] Son Luu Nguyen [email protected] Du Huu Nguyen [email protected] 1
Department of Applied Mathematics, Faculty of Applied Science, University of Technology, Vietnam National University Ho Chi Minh City, 268 Ly Thuong Kiet, District 10, Ho Chi Minh City, Vietnam
2
Department of Mathematics, University of Puerto Rico, Rio Piedras Campus, San Juan, PR 00936, USA
3
Department of Mathematics, Mechanics and Informatics, Hanoi National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam
D.T. Nguyen et al.
fields such as chemistry, biology, game theory, and finance. When the number of particles increases, the analysis and computation of the equations becomes more and more complicated and time consuming. One way to overcome this challenge is replacing all interactions with particles by a single average interaction, which is normally represented by an empirical measure associated to a system. Studying the limits of mean-field models as the sizes of the systems tend to infinity has arisen from modelling in science and engineering with many technical difficulties (see, for example, [4, 10, 16, 19, 22, 23]). In recent decades, many studies of mean-field models with different settings have been done such as models with space noises [9, 15], models with a common noise [6, 14], models with jumps [1], regime-switching models [18, 24], and models with multi-classes [17]. In these works, the authors investigated the limits of the empirical measures of various mean field systems as the number of particles tends to infinity under the global Lipschitz and linear growth conditions. Although global Lip
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