Poisson equation on Wasserstein space and diffusion approximations for multiscale McKean-Vlasov equation. (English) Zbl 1534.60106
Summary: We consider the fully-coupled McKean-Vlasov equation with multi-time-scale potentials, and all the coefficients depend on the distributions of both the slow component and the fast motion. By studying the smoothness of the solution of the Poisson equation on Wasserstein space, we derive the asymptotic limit as well as the quantitative error estimate of the convergence for the slow process. An extra homogenized drift term containing derivative in the measure argument of the solution of the Poisson equation appears in the limit, which seems to be new and is unique for systems involving the fast distribution.
MSC:
| 60J60 | Diffusion processes |
| 60F05 | Central limit and other weak theorems |
| 35J60 | Nonlinear elliptic equations |
| 70K70 | Systems with slow and fast motions for nonlinear problems in mechanics |
Keywords:
Poisson equation on Wasserstein space; diffusion approximation; McKean-Vlasov equation; multiscale processesReferences:
| [1] | Armstrong, S., Cardaliaguet, P., and Souganidis, P., Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations, J. Amer. Math. Soc., 27 (2014), pp. 479-540. · Zbl 1286.35023 |
| [2] | Abdulle, A., Pavliotis, G. A., and Vaes, U., Spectral methods for multiscale stochastic differential equations, SIAM/ASA J. Uncertain. Quantif., 5 (2017), pp. 720-761. · Zbl 1398.65313 |
| [3] | Bao, J., Yin, G., and Yuan, C., Two-time-scale stochastic partial differential equations driven by \(\alpha \)-stable noises: Averaging principles, Bernoulli, 23 (2018), pp. 645-669. · Zbl 1360.60118 |
| [4] | Barbu, V. and Röckner, M., From non-linear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Probab., 48 (2020), pp. 1902-1920. · Zbl 1469.60216 |
| [5] | Barbu, V., Röckner, M., and Russo, F., Probabilistic representation for solutions of an irregular porous media type equation: The degenerate case, Probab. Theory Relat. Fields, 15 (2011), pp. 1-43. · Zbl 1227.60088 |
| [6] | Bezemek, Z. W. and Spiliopoulos, K., Large deviations for interacting multiscale particle systems, Stochastic Process. Appl., 155 (2023), pp. 27-108. · Zbl 1508.60036 |
| [7] | Bezemek, Z. W. and Spiliopoulos, K., Rate of homogenization for fully-coupled McKean-Vlasov SDEs, Stoch. Dyn., 23 (2023), 2350013. · Zbl 1521.60035 |
| [8] | Bezemek, Z. W. and Spiliopoulos, K., Moderate deviations for fully coupled multiscale weakly interacting particle systems, Stoch. Partial Differ. Equ. Anal. Comput., (2023), doi:10.1007/s40072-023-00301-0. · Zbl 07880267 |
| [9] | Bogachev, V. I., Shaposhnikov, S. V., and Veretennikov, A. Yu., Differentiability of solutions of stationary Fokker-Planck-Kolmogorov equations with respect to a parameter, Discrete Contin. Dyn. Syst., 36 (2016), pp. 3519-3543. · Zbl 1346.60099 |
| [10] | Buckdahn, R., Li, J., Peng, S., and Rainer, C., Mean-field stochastic differential equations and associated PDEs, Ann. Probab., 45 (2017), pp. 824-787. · Zbl 1402.60070 |
| [11] | Cardaliaguet, P., Notes on mean field games, from Lectures at Collége de France, Paris, 2013. |
| [12] | Carmona, R. and Delarue, F., Probabilistic Theory of Mean Field Games with Applications I: Mean Field FBSDEs, Control, and Games, , Springer, 2018. · Zbl 1422.91014 |
| [13] | Carrillo, J. A. and Choi, Y. P., Quantitative error estimates for the large friction limit of Vlasov equation with nonlocal forces, Ann. Inst. H. Poincaré Anal. Non Linéaire, 37 (2020), pp. 925-954. · Zbl 1440.35322 |
| [14] | Carrillo, J. A., Gvalani, R. S., Pavliotis, G. A., and Schlichting, A., Long-time behaviour and phase transitions for the Mckean-Vlasov equation on the torus, Arch. Ration. Mech. Anal., 235 (2020), pp. 635-690. · Zbl 1436.58025 |
| [15] | Chaudru de Raynal, P.-E. and Frikha, N., Well-Posedness for some non-linear SDEs and related PDE on the Wassertein space, J. Math. Pures Appl., 159 (2022), pp. 1-167. · Zbl 1494.60063 |
| [16] | Chen, X., Chen, Z.-Q., Kumagai, T., and Wang, J., Periodic homogenization of non-symmetric Lévy-type processes, Ann. Probab., 49 (2021), pp. 2874-2921. · Zbl 1486.60061 |
| [17] | Cerrai, S. and Lunardi, A., Averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations: The almost periodic case, SIAM J. Math. Anal., 49 (2017), pp. 2843-2884. · Zbl 1370.60102 |
| [18] | Crisan, D. and McMurray, E., Smoothing properties of McKean-Vlasov SDEs, Probab. Theory Relat. Fields, 171 (2018), pp. 97-148. · Zbl 1393.60074 |
| [19] | Dawson, D. A., Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys., 31 (1983), pp. 29-85. |
| [20] | Delgadino, M. G., Gvalani, R. S., and Pavliotis, G. A., On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions, Arch. Ration. Mech. Anal., 241 (2021), pp. 91-148. · Zbl 07364831 |
| [21] | Delgadino, M. G., Gvalani, R. S., Pavliotis, G. A., and Smith, S. A., Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions, Comm. Math. Phys., 401 (2023), pp. 275-323. · Zbl 07700884 |
| [22] | Duerinckx, M., Gloria, A., and Otto, F., The structure of fluctuations in stochastic homogenization, Comm. Math. Phys., 377 (2020), pp. 259-306. · Zbl 1441.35026 |
| [23] | Duncan, A. B., Duong, M. H., and Pavliotis, G. A., Brownian motion in an \(N\)-scale periodic potential, J. Stat. Phys., 190 (2023), 82. · Zbl 1512.35039 |
| [24] | Eberle, A., Guillin, A., and Zimmer, R., Quantitative Harris type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc., 371 (2019), pp. 7135-7173. · Zbl 1481.60154 |
| [25] | Fournier, N. and Guillin, A., From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Éc. Norm. Supér., 50 (2017), pp. 157-199. · Zbl 1371.82086 |
| [26] | Gomes, S. N. and Pavliotis, G. A., Mean field limits for interacting diffusions in a two-scale potential, J. Nonlinear Sci., 28 (2018), pp. 905-941. · Zbl 1394.35494 |
| [27] | Gvalania, R. S. and Schlichting, A., Barriers of the McKean-Vlasov energy via a mountain pass theorem in the space of probability measures, J. Funct. Anal., 279 (2020), 108720. · Zbl 1473.60061 |
| [28] | Hairer, M. and Pardoux, E., Fluctuations around a homogenised semilinear random PDE, Arch. Ration. Mech. Anal., 239 (2021), pp. 151-217. · Zbl 1456.35243 |
| [29] | Helmes, K., Stockbridge, R., and Zhu, C., Continuous inventory models of diffusion type: Long-term average cost criterion, Ann. Appl. Probab., 27 (2017), pp. 1831-1885. · Zbl 1369.93720 |
| [30] | Hong, W., Li, S., and Liu, W., Strong convergence rates in averaging principle for slow-fast McKean-Vlasov SPDEs, J. Differential Equations, 316 (2022), pp. 94-135. · Zbl 1487.60123 |
| [31] | Honoré, I., Menozzi, S., and Pagès, G., Non-asymptotic Gaussian estimates for the recursive approximation of the invariant measure of a diffusion, Ann. Inst. Henri Poincaré Probab. Stat., 56 (2020), pp. 1559-1605. · Zbl 1472.60112 |
| [32] | Huang, X. and Wang, F.-Y., Distribution dependent SDEs with singular coefficients, Stochastic Process. Appl., 129 (2019), pp. 4747-4770. · Zbl 1433.60034 |
| [33] | Jabin, P.-E., Macroscopic limit of Vlasov type equations with friction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), pp. 651-672. · Zbl 0965.35013 |
| [34] | Kac, M., Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability: Contributions to Astronomy and Physics, , University of California Press, Berkeley, CA, 1956, pp. 171-197. · Zbl 0072.42802 |
| [35] | Khasminskii, R. Z. and Yin, G., On averaging principles: An asymptotic expansion approach, SIAM J. Math. Anal., 35 (2004), pp. 1534-1560. · Zbl 1072.34054 |
| [36] | Khasminskii, R. Z. and Yin, G., Limit behavior of two-time-scale diffusions revised, J. Differential Equations, 212 (2005), pp. 85-113. · Zbl 1112.35014 |
| [37] | Lions, P., Mean-field Games and Applications, Lectures at Collége de France, Paris, 2007. |
| [38] | Mattingly, J. C., Stuart, A. M., and Tretyakov, M. V., Convergence of numerical time-averaging and stationary measures via Poisson equations, SIAM J. Numer. Anal., 48 (2010), pp. 552-577. · Zbl 1217.65014 |
| [39] | McKean, H. P. Jr., A class of Markov processes associated with nonlinear parabolic equations, Proc. Nat. Acad. Sci. USA, 56 (1966), pp. 1907-1911. · Zbl 0149.13501 |
| [40] | Mishura, Y. S. and Veretennikov, A. Yu., Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, Theory Probab. Math. Statist., 103 (2021), pp. 59-101. · Zbl 1482.60079 |
| [41] | Pagès, G. and Panloup, F., Ergodic approximation of the distribution of a stationary diffusion: Rate of convergence, Ann. Appl. Probab., 22 (2012), pp. 1059-1100. · Zbl 1252.60080 |
| [42] | Pardoux, E. and Veretennikov, A. Yu., On the Poisson equation and diffusion approximation. I, Ann. Probab., 29 (2001), pp. 1061-1085. · Zbl 1029.60053 |
| [43] | Pardoux, E. and Veretennikov, A. Yu., On the Poisson equation and diffusion approximation 2, Ann. Probab., 31 (2003), pp. 1166-1192. · Zbl 1054.60064 |
| [44] | Pardoux, E. and Veretennikov, A. Yu., On the Poisson equation and diffusion approximation 3, Ann. Probab., 33 (2005), pp. 1111-1133. · Zbl 1071.60022 |
| [45] | Röckner, M., Sun, X., and Xie, Y. C., Strong convergence order for slow-fast McKean-Vlasov stochastic differential equations, Ann. Inst. Henri Poincaré Probab. Statist., 57 (2021), pp. 547-576. · Zbl 1491.60088 |
| [46] | Wang, F.-Y., Distribution dependent SDEs for Landau type equations, Stochastic Process. Appl., 128 (2018), pp. 595-621. · Zbl 1380.60077 |
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