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On continuity equations in infinite dimensions with non-Gaussian reference measure. (English) Zbl 1317.60083
The authors study infinite-dimensional equations of the form $$\dot{\mu}+ \text{div}(\mu\cdot b) = 0$$, $$\mu_0=\zeta$$, where $$\mu=\mu_t(dx)$$ is a curve of probability measures on $$\mathbb R^{\infty}$$, equipped with the product $$\sigma$$-algebra induced by the Borel $$\sigma$$-algebra on $$\mathbb R$$ and $$b$$ is the function from $$\mathbb R^{\infty}$$ to $$\mathbb R^{\infty}$$. The approach to solve this equation is to choose a reference measure $$\nu$$ and search for solutions which are of the form $$\mu_t(dx)=\rho(t,x)\cdot\nu(dx)$$, where $$\rho$$ is the solution of the equation $$\dot{\rho} +\text{div}_{\nu}(\rho\cdot b) = 0$$, $$\rho(0, x) = \rho_0$$, with $$\zeta=\rho_0\cdot \nu$$. The choice of reference measure is at author’s disposal and depends on $$b$$. One usually takes a Gaussian measure as reference measure, since they are best studied. However, in many cases this is not the best choice. The key point is to choose the reference measure in such a way that it will be possible to formulate existence and uniqueness conditions. It is possible due to the fact that many probability measures on $$\mathbb R^d$$ are images of Gaussian measures under so-called triangular mappings which turn out to have sufficient regularity. Therefore, one can reduce existence and uniqueness questions to the case of a Gaussian reference measure. The uniqueness of the solution is a more difficult problem compared to the existence. An existence result is established for the case of reference measures $$\nu$$ on $$\mathbb R^{\infty}$$ with logarithmic derivatives integrable in any power. Uniqueness is proved for a wide class of product measures, including log-concave ones. Another uniqueness result is proved for a class of uniformly log-concave Gibbs measures.
The main approach relies on the mass transportation method. More precisely, triangular mass transportation is chosen. Sobolev estimates for triangular mappings are established and the key technical relations between transport equations and mass transfer are established as an auxiliary result.

##### MSC:
 60H30 Applications of stochastic analysis (to PDEs, etc.) 35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) 35F10 Initial value problems for linear first-order PDEs 35B45 A priori estimates in context of PDEs 35L45 Initial value problems for first-order hyperbolic systems 46E27 Spaces of measures 58D20 Measures (Gaussian, cylindrical, etc.) on manifolds of maps
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##### References:
  Albeverio, S.; Kondratiev, Yu. G.; Röckner, M.; Tsikalenko, T. V., Uniqueness of Gibbs states for quantum lattice systems, Probab. Theory Related Fields, 108, 193-218, (1997) · Zbl 0883.60094  Albeverio, S.; Kondratiev, Yu. G.; Röckner, M.; Tsikalenko, T. V., A-priori estimates for symmetrizing measures and their applications to Gibbs states, J. Funct. Anal., 171, 366-400, (2000) · Zbl 0968.60095  Albeverio, S.; Röckner, M., Classical Dirichlet forms on topological vector spaces. closability and a cameron-martin formula, J. Funct. Anal., 88, 395-436, (1990) · Zbl 0737.46036  Ambrosio, L., Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158, 227-260, (2004) · Zbl 1075.35087  Ambrosio, L., Transport equations and Cauchy vector problems for non-smooth vector fields, (Lecture Notes of the CIME Summer School in Cetrary, (June 27-July 2, 2005))  Ambrosio, L.; Figalli, A., On flows associated to Sobolev vector fields in Wiener spaces: an approach a la diperna-Lions, J. Funct. Anal., 256, 1, 179-214, (2009) · Zbl 1156.60036  Ambrosio, L.; Figalli, A., Almost everywhere well-posedness of continuity equations with measure initial data, C. R. Math., 348, 5-6, 249-252, (2010) · Zbl 1195.35202  Bogachev, V. I., Measure theory, vols. 1, 2, (2007), Springer Berlin-New York  Bogachev, V. I., Differentiable measures and Malliavin calculus, Math. Surveys Monogr., vol. 164, (2010), AMS · Zbl 1247.28001  Bogachev, V. I.; Da Prato, G.; Röckner, M.; Shaposhnikov, S. V., Nonlinear evolution equations for measures on infinite dimensional spaces, (Stochastic Part. Diff. Equations and Appl., Quad. Mat., vol. 25, (2010)), 51-64 · Zbl 1275.60051  Bogachev, V. I.; Kolesnikov, A. V., On the Monge-ampere equation in infinite dimensions, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8, 4, 547-572, (2005) · Zbl 1103.49001  Bogachev, V. I.; Kolesnikov, A. V., On nonlinear transformation of convex measures, Theory Probab. Appl., 50, 1, 34-52, (2005) · Zbl 1091.28009  Bogachev, V. I.; Kolesnikov, A. V., The Monge-Kantorovich problem: achievements, connections, and perspectives, Russian Math. Surveys, 67, 5, 785-890, (2012) · Zbl 1276.28029  Bogachev, V. I.; Kolesnikov, A. V., Sobolev regularity for the Monge-ampere equation in the Wiener space, Kyoto J. Math., 53, 4, 713-890, (2013) · Zbl 1286.28011  Bogachev, V. I.; Kolesnikov, A. V.; Medvedev, K. V., Triangular transformations of measures, Sb. Math., 196, 3, 3-30, (2005) · Zbl 1072.28010  Bogachev, V. I.; Mayer-Wolf, E., Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions, J. Funct. Anal., 167, 1-68, (1999) · Zbl 0956.60079  Caffarelli, L. A., Interior $$W^{2, p}$$-estimates for solutions of the Monge-Ampère equation, Ann. of Math. (2), 131, 1, 135-150, (1990) · Zbl 0704.35044  Caffarelli, L. A., Monotonicity properties of optimal transportation and the FKG and related inequalities, Comm. Math. Phys., 214, 3, 547-563, (2000) · Zbl 0978.60107  Cruzeiro, A. B., Èquations différentielles sur l’espace de Wiener et formules de cameron-martin non linéaires, J. Funct. Anal., 54, 206-227, (1983) · Zbl 0524.47028  Di Perna, R. J.; Lions, P. L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547, (1989) · Zbl 0696.34049  Di Perna, R. J.; Lions, P. L., On the Cauchy problem for the Boltzmann equation: global existence and weak stability, Ann. of Math., 130, 312-366, (1989)  Fang, S.; Luo, D., Transport equations and quasi-invariant flows on the Wiener space, Bull. Sci. Math., 134, 295-328, (2010) · Zbl 1201.60053  Feyel, D.; Üstünel, A. S., Monge-Kantorovich measure transportation and Monge-Ampère equation on Wiener spaces, Probab. Theory Related Fields, 128, 347-385, (2004) · Zbl 1055.60052  Kolesnikov, A. V., Convexity inequalities and optimal transport of infinite-dimensional measures, J. Math. Pures Appl. (9), 83, 11, 1373-1404, (2004) · Zbl 1087.49034  Kolesnikov, A. V., Mass transportation and contractions, MIPT Proc., 2, 4, 90-99, (2010)  Kolesnikov, A. V., On global Hölder estimates for optimal transportation, Mat. Zametki, 88, 5, 708-728, (2010)  Kolesnikov, A. V., On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57, 2, 296-321, (2012)  Kolesnikov, A. V.; Tikhonov, S. Yu., Regularity of the Monge-Ampère equation in Besov’s spaces, Calc. Var. Partial Differential Equations, (2014), in press · Zbl 1290.35098  Kolesnikov, A. V.; Zaev, D., Optimal transportation of processes with infinite Kantorovich distance. independence and symmetry · Zbl 1369.49066  Le Bris, C.; Lions, P. L., Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33, 7, 1272-1317, (2007) · Zbl 1157.35301  Nualart, D., The Malliavin calculus and related topics, (2006), Springer-Verlag Berlin Heidelberg · Zbl 1099.60003  Ovsienko, Yu.; Zhdanov, R., Estimates for Sobolev norms of triangular mappings, Moscow Univ. Math. Bull., 62, 1, 1-4, (2010)  Villani, C., Topics in optimal transportation, (2003), Amer. Math. Soc. Providence, RI · Zbl 1106.90001
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