Abstract
The Kantorovich problem with the cost function given by the Cameron–Martin norm is considered for nonlinear images of the Wiener measure that are distributions of one-dimensional diffusion processes with nonconstant diffusion coefficients. It is shown that the problem can have trivial solutions only if the derivative of the diffusion coefficient differs from zero almost everywhere.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
D. B. Bukin, “On theMonge and Kantorovich problems for distributions of diffusion processes,” Math. Notes 96 (5–6), 864–870 (2014).
V. I. Bogachev, Gaussian Measures, in Math. Surveys Monogr. (Amer. Math. Soc., Providence, RI, 1998), Vol. 62.
V. I. Bogachev, Differentiable Measures and the Malliavin Calculus, in Math. Surveys Monogr. (Amer. Math. Soc., Providence, RI, 2010), Vol. 164.
V. I. Bogachev, “Gaussian measures on infinite-dimensional spaces,” in Real and Stochastic Analysis. Current Trends (World Sci. Publ., Hackensack, NJ, 2014), pp. 1–83.
D. Feyel and A. S. Üstünel, “Monge–Kantorovitch measure transportation and Monge–Ampère equation on Wiener space,” Probab. Theory Related Fields 128 (3), 347–385 (2004).
D. Feyel and A. S. Üstünel, “Solution of the Monge–Ampère equation on Wiener space for general log-concave measures,” J. Funct. Anal. 232 (1), 29–55 (2006).
V. I. Bogachev and A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and prospects,” Uspekhi Mat. Nauk 67 (5 (407)), 3–110 (2012) [Russian Math. Surveys 67 (5), 785–890 (2012)].
V. I. Bogachev and A. V. Kolesnikov, “On the Monge–Ampère equation in infinite dimensions,” Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (4), 547–572 (2005).
V. I. Bogachev and A. V. Kolesnikov, “Sobolev regularity for the Monge–Ampère equation in the Wiener space,” Kyoto J. Math. 53 (4), 713–738 (2013).
F. Cavalletti, “The Monge problem in Wiener space,” Calc. Var. Partial Differential Equations 45 (1–2), 101–124 (2012).
S. Fang, J. Shao, and K. -Th. Sturm, “Wasserstein space over the Wiener space,” Probab. Theory Related Fields 146 (3–4), 535–565 (2010).
V. I. Bogachev, A. V. Kolesnikov, and K. V. Medvedev, “Triangular transformations of measures,” Mat. Sb. 196 (3), 3–30 (2005) [Sb. Math. 196 (3), 309–335 (2005)].
I. I. Gihman and A. V. Skorohod, The Theory of Stochastic Processes, (Nauka, Moscow, 1975; Springer-Verlag, Berlin–New York, 1979), Vol. III.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes (North-Holland Publishing Co., Amsterdam–New York, Kodansha Ltd, Tokyo, 1981; Nauka, Moscow, 1986).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © D. B. Bukin, 2016, published in Matematicheskie Zametki, 2016, Vol. 100, No. 5, pp. 682–688.
Rights and permissions
About this article
Cite this article
Bukin, D.B. On the Kantorovich problem for nonlinear images of the Wiener measure. Math Notes 100, 660–665 (2016). https://doi.org/10.1134/S000143461611002X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143461611002X