Abstract
We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.
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Ambrosio L., Gigli N., Savaré G.: Gradient flows in metric spaces and in the space of probability measures. Birkhäuser, Basel (2005)
Ambrosio L., Kirchheim B., Pratelli A.: Existence of optimal transport maps for crystalline norms. Duke Math. J. 125(2), 207–241 (2004)
Ambrosio, L., Pratelli, A.: Existence and stability results in the L 1 theory of optimal transportation. In: Optimal Transportation and Applications, volume 1813 of Lecture notes in mathematics. Springer, Berlin (2001)
Ambrosio L., Tilli P.: Topics on analysis in metric spaces. Oxford Lecture series in Mathematics and its application. Oxford University Press, Oxford (2004)
Bianchini S., Caravenna L.: On the extremality, uniqueness and optimality of transference plans. Bull. Inst. Math. Acad. Sin. (N.S.) 4(4), 353–454 (2009)
Bianchini, S., Cavalletti, F.: The Monge problem for distance cost in geodesic spaces. preprint SISSA 50/2009/M, arXiv:1103.2796 (2009)
Bogachev, V.I.: Gaussian Measures. Mathematical surveys and monographs. AMS (1998)
Caffarelli L., Feldman M., McCann R.J.: Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Am. Math. Soc. 15, 1–26 (2002)
Caravenna, L.: An existence result for the Monge problem in \({\mathbb{R}^n}\) with norm cost function. preprint (2009)
Cavalletti, F.: Optimal transport with branching distance cost and the obstacle problem. SIAM J. Math. Anal. (2011, to appear)
Champion T., De Pascale L.: The Monge problem in \({\mathbb{R}^d}\) . Duke Math. J. 157(3), 551–572 (2011)
Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich mass transfer problem. In: Current Developments in Mathematics, pp. 65–126. International Press, Boston (1997)
Feldman M., McCann R.: Monge’s transport problem on a Riemannian manifold. Trans. Am. Math. Soc. 354, 1667–1697 (2002)
Feyel D., A.S.: Monge-Kantorovich measure transportation and Monge-Ampére equation on Wiener space. Probab. Theory Relat. Fields 128, 347–385 (2004)
Fremlin, D.H.: Measure Theory, vol. 4. Torres Fremlin, Colchester (2002)
Larman D.G.: A compact set of disjoint line segments in \({\mathbb{R}^{3}}\) whose end set has positive measure. Mathematika 18, 112–125 (1971)
Srivastava, A.M.: A Course on Borel Sets. Springer, Berlin (1998)
Sudakov V.N.: Geometric problems in the theory of dimensional distributions. Proc. Steklov Inst. Math. 141, 1–178 (1979)
Trudinger N., Wang X.J.: On the Monge mass transfer problem. Calc. Var. Partial Differ. Equ. 13, 19–31 (2001)
Villani, C.: Optimal Transport, Old and New. Springer, Berlin (2008)
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Communicated by L. Ambrosio.