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Distances between transition probabilities of diffusions and applications to nonlinear Fokker-Planck-Kolmogorov equations. (English) Zbl 1345.35051
The authors consider the Cauchy problem for measures \[ \begin{aligned} \partial_t\mu &=\mathcal{L}_{A,b}^*\mu, \\ \mu|_{t=0}&=\nu,\end{aligned}\tag{1} \] in \(\mathbb{R}^d\times (0,T)\), where \(\nu\) is a Borel probability measure on \(\mathbb{R}^d\) and the coefficients of the operator \[ \mathcal{L}_{A,b}=\sum_{i,j=1}^d a^{ij}\partial_{x_i}\partial_{x_j}+\sum_{i=1}^db^i\partial_{x_i} \] are Borel-measurable. The diffusion matrix \(A(x,t)=(a^{ij}(x,t))_{ij}\) is positive symmetric, assumed to be locally Lipschitz in \(x\) and locally strictly positive, and the drift \(b(x,t)=(b^i(x,t))_i\) is locally bounded.
The main result of the paper are estimates for the entropy of two solutions of the Cauchy problem \((1)\) with different diffusion matrices and drifts (but the same initial data \(\nu\)). More precisely, let \(A^{(i)}\) and \(b^{(i)}\) be diffusion matrices and drift functions satisfying the above conditions, and \(\mu^{(i)}\) be the corresponding solutions of \((1)\) with densities \(\rho^{(i)}\), \(i=1,2.\) If either the coefficients of elliptic operator satisfy some mild integrability conditions, or a certain Lyapunov function exists, estimates for the entropy \[ H(\mu_t^{(1)}|\mu_t^{(2)})= \int_{\mathbb{R}^d}\rho^{(1)}(x, t)\log\Big(\frac{\rho^{(1)}(x, t)}{\rho^{(2)}(x, t)}\Big)dx \] are established, and as a consequence, bounds for the total variation and Kantorovich distance between \(\rho^{(1)}\) and \(\rho^{(2)}\) deduced. The article is based on a series of previous works by the authors, the main novelty is to consider equations with different diffusion matrices (and drifts).
Finally, the results are applied to prove the local existence and uniqueness of solutions of nonlinear Fokker-Planck-Kolmogorov equations and to formulate sufficient conditions for their differentiability. Further applications are an optimal control problem and mean field game models.

MSC:
35K15 Initial value problems for second-order parabolic equations
60J60 Diffusion processes
35Q84 Fokker-Planck equations
35K55 Nonlinear parabolic equations
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[1] Ambrosio, L.; Savaré, G.; Zambotti, L., Existence and stability for Fokker-Planck equations with log-concave reference measure, Probab. Theory Related Fields, 145, 3-4, 517-564, (2009) · Zbl 1235.60105
[2] Annunziato, M.; Borzi, A., Optimal control of probability density functions of stochastic processes, Math. Model. Anal., 15, 4, 393-407, (2010) · Zbl 1216.35154
[3] Annunziato, M.; Borzi, A., A Fokker-Planck control framework for multidimensional stochastic processes, J. Comput. Appl. Math., 237, 487-507, (2013) · Zbl 1251.35196
[4] Arnold, A.; Markowich, P.; Toscani, G.; Unterreiter, A., On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations, Comm. Partial Differential Equations, 26, 1-2, 43-100, (2001) · Zbl 0982.35113
[5] Bensoussan, A.; Frehse, J.; Yam, Ph., Mean field games and mean field type control theory, (2013), Springer New York · Zbl 1287.93002
[6] Bogachev, V. I., Measure theory, vols. 1, 2, (2007), Springer Berlin-New York
[7] Bogachev, V. I., Differentiable measures and the Malliavin calculus, (2010), Amer. Math. Soc. Providence, Rhode Island · Zbl 0929.58015
[8] Bogachev, V. I.; Da Prato, G.; Röckner, M., On parabolic equations for measures, Comm. Partial Differential Equations, 33, 1-3, 397-418, (2008) · Zbl 1323.35058
[9] Bogachev, V. I.; Kirillov, A. I.; Shaposhnikov, S. V., The Kantorovich and variation distances between invariant measures of diffusions and nonlinear stationary Fokker-Planck-Kolmogorov equations, Math. Notes, 96, 6, 17-25, (2014) · Zbl 1315.35221
[10] Bogachev, V. I.; Kolesnikov, A. V., The Monge-Kantorovich problem: achievements, connections, and perspectives, Uspekhi Mat. Nauk, Russian Math. Surveys, 67, 5, 785-890, (2012), (in Russian); English transl.: · Zbl 1276.28029
[11] Bogachev, V. I.; Krylov, N. V.; Röckner, M., On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations, 26, 11-12, 2037-2080, (2001) · Zbl 0997.35012
[12] Bogachev, V. I.; Krylov, N. V.; Röckner, M., Elliptic and parabolic equations for measures, Russian Math. Surveys, 64, 6, 973-1078, (2009) · Zbl 1194.35481
[13] Bogachev, V. I.; Krylov, N. V.; Röckner, M.; Shaposhnikov, S. V., Fokker-Planck-Kolmogorov equations, (2015), Amer. Math. Soc. Rhode Island, Providence · Zbl 1321.35242
[14] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., Global regularity and bounds for solutions of parabolic equations for probability measures, Teor. Veroyatn. Primen., Theory Probab. Appl., 50, 4, 561-581, (2006), (in Russian); English transl.: · Zbl 1203.60095
[15] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., Estimates of densities of stationary distributions and transition probabilities of diffusion processes, Teor. Veroyatn. Primen., Theory Probab. Appl., 52, 2, 209-236, (2008), (in Russian); English transl.: · Zbl 1154.35322
[16] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., Nonlinear evolution and transport equations for measures, Dokl. Ross. Akad. Nauk, Dokl. Math., 80, 3, 785-789, (2009), (in Russian); English transl.: · Zbl 1200.35047
[17] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., On uniqueness problems related to elliptic equations for measures, J. Math. Sci. (N. Y.), 176, 6, 759-773, (2011) · Zbl 1290.35289
[18] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., On uniqueness problems related to the Fokker-Planck-Kolmogorov equation for measures, J. Math. Sci. (N. Y.), 179, 1, 7-47, (2011) · Zbl 1291.35425
[19] Bogachev, V. I.; Shaposhnikov, S. V.; Veretennikov, A. Yu., Differentiability of solutions of stationary Fokker-Planck-Kolmogorov equations with respect to a parameter, Discrete Contin. Dyn. Syst. A, 36, 7, 3519-3543, (2016) · Zbl 1346.60099
[20] Bogachev, V. I.; Veretennikov, A. Yu.; Shaposhnikov, S. V., Differentiability of invariant measures of diffusions with respect to a parameter, Dokl. Akad. Nauk, Dokl. Math., 91, 1, 76-79, (2015), (in Russian); English transl.: · Zbl 1325.35226
[21] Bolley, F.; Gentil, I., Phi-entropy inequalities for diffusion semigroups, J. Math. Pures Appl., 93, 5, 449-473, (2010) · Zbl 1193.47046
[22] Bolley, F.; Gentil, I.; Guillin, A., Dimensional contraction via Markov transportation distance, J. Lond. Math. Soc. (2), 90, 1, 309-332, (2014) · Zbl 1312.47055
[23] Bolley, F.; Guillin, A.; Villani, C., Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137, 3-4, 541-593, (2007) · Zbl 1113.60093
[24] Bolley, F.; Villani, C., Weighted csiszar-Kullback-Pinsker inequalities and applications to transportation inequalities, Ann. Fac. Sci. Toulouse Math. (6), 14, 3, 331-352, (2005) · Zbl 1087.60008
[25] Carrillo, J. A.; McCann, R. J.; Villani, C., Kinetic equilibration rates for granular media and related equations: entropy, dissipation and mass transportation estimates, Rev. Mat. Iberoam., 19, 971-1018, (2003) · Zbl 1073.35127
[26] Carrillo, J. A.; Toscani, G., Exponential convergence toward equilibrium for homogeneous Fokker-Planck-type equations, Math. Methods Appl. Sci., 21, 13, 1269-1286, (1998) · Zbl 0922.35131
[27] Dobrushin, R. L., Vlasov equations, Funktsional. Anal. i Prilozhen., Funct. Anal. Appl., 13, 2, 115-123, (1979), (in Russian); English transl.: · Zbl 0422.35068
[28] Dolbeault, J.; Nazaret, B.; Savaré, G., From Poincaré to logarithmic Sobolev inequalities: a gradient flow approach, SIAM J. Math. Anal., 44, 5, 3186-3216, (2012) · Zbl 1264.26011
[29] Gomes, D. A.; Saude, J., Mean field games models — a brief survey, Dyn. Games Appl., 4, 110-154, (2014) · Zbl 1314.91048
[30] Gozlan, N., Integral criteria for transportation-cost inequalities, Electron. Commun. Probab., 11, 64-77, (2006) · Zbl 1112.60009
[31] Guéant, O.; Lasry, J.-M.; Lions, P.-L., Mean field games and applications, (Lecture Notes in Math., vol. 2003, (2011)), 205-266
[32] Kabanov, Yu. M.; Liptser, R. Sh.; Shiryaev, A. N., On the variation distance for probability measures defined on a filtered space, Probab. Theory Related Fields, 71, 1, 19-35, (1986) · Zbl 0554.60006
[33] Krylov, N. V.; Priola, E., Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35, 1, 1-22, (2010) · Zbl 1195.35160
[34] Liese, F., Hellinger integrals of diffusion processes, Statistics, 17, 1, 63-78, (1986) · Zbl 0598.60042
[35] Liese, F.; Schmidt, W., A note on the convergence of integral functionals of diffusion processes. an application to strong convergence, Math. Nachr., 161, 283-289, (1993) · Zbl 0795.60048
[36] Liese, F.; Schmidt, W., On the strong convergence, contiguity and entire separation of diffusion processes, Stoch. Stoch. Rep., 50, 3-4, 185-203, (1994) · Zbl 0832.60064
[37] Lions, J. L., Optimal control of systems governed by partial differential equations, (1971), Springer Berlin · Zbl 0203.09001
[38] Manita, O. A., Estimates for the Kantorovich distances between solutions to the nonlinear Fokker-Planck-Kolmogorov equation with monotone drift, (2015)
[39] Manita, O. A.; Romanov, M. S.; Shaposhnikov, S. V., On uniqueness of solutions to nonlinear Fokker-Planck-Kolmogorov equations, Nonlinear Anal., 128, 199-226, (2015) · Zbl 1336.35334
[40] Manita, O. A.; Shaposhnikov, S. V., Nonlinear parabolic equations for measures, Algebra i Analiz, St. Petersburg Math. J., 25, 1, 43-62, (2014), (in Russian); English transl.: · Zbl 1286.35137
[41] Mohammadi, M.; Borzi, A., A Hermite spectral method for a Fokker-Planck optimal control problem in an unbounded domain, Int. J. Uncertain. Quantif., 5, 3, 233-254, (2015)
[42] Natile, L.; Peletier, M. A.; Savaré, G., Contraction of general transportation costs along solutions to Fokker-Planck equations with monotone drifts, J. Math. Pures Appl. (9), 95, 1, 18-35, (2011) · Zbl 1206.35236
[43] Otto, F.; Westdickenberg, M., Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal., 37, 1227-1255, (2005) · Zbl 1094.58016
[44] Pardoux, E.; Veretennikov, A. Yu., On the Poisson equation and diffusion approximation. II, Ann. Probab., 31, 3, 1166-1192, (2003) · Zbl 1054.60064
[45] von Renesse, M.-K.; Sturm, K.-T., Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math., 68, 923-940, (2005) · Zbl 1078.53028
[46] Shaposhnikov, S. V., On the uniqueness of the probabilistic solution of the Cauchy problem for the Fokker-Planck-Kolmogorov equation, Teor. Veroyatn. Primen., Theory Probab. Appl., 56, 1, 96-115, (2012), (in Russian); English transl.: · Zbl 1238.35168
[47] Shaposhnikov, S. V., The Fokker-Planck-Kolmogorov equations with a potential and a non-uniformly elliptic diffusion matrix, Tr. Mosk. Mat. Obs., Trans. Moscow Math. Soc., 74, 1, 15-29, (2013), (in Russian); English transl.: · Zbl 1310.35229
[48] Veretennikov, A. Yu., On Sobolev solutions of Poisson equations in \(\mathbb{R}^d\) with a parameter, J. Math. Sci. (N. Y.), 179, 1, 48-79, (2011) · Zbl 1291.35042
[49] Villani, C., Optimal transport. old and new, (2009), Springer-Verlag Berlin · Zbl 1156.53003
[50] Wang, F.-Y., From super Poincaré to weighted log-Sobolev and entropy-cost inequalities, J. Math. Pures Appl., 90, 3, 270-285, (2008) · Zbl 1155.60009
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