##
**Transportation cost-information inequalities and applications to random dynamical systems and diffusions.**
*(English)*
Zbl 1061.60011

Let \((E,d)\) be a metric space equipped with a \(\sigma\)-algebra \(\mathcal{B}\) such that the metric is \(\mathcal{B}\times \mathcal{B}\)-measurable. A probability \(\mu\) satisfies the \(L^p\)-transportation cost information inequality if
\[
W_p^d(\mu,\nu) \leq \sqrt{2 C H(\nu| \mu)} \quad \text{in short: } \mu\in T_p(C)
\]
for some constant \(C>0\), where the Wasserstein distance \(W_p^d\) is defined as
\[
W_p^d(\mu,\nu) := \inf_{\pi} \left\{ \int_{E\times E} d^p d\pi : \pi\in M^1(E\times E) \;\text{with marginals } \;\mu, \nu \right\},
\]
and the Kullback information \(H\) is defined as \(\int \log \frac{d\nu}{d\mu} d\nu \) (if \(\nu \ll \mu\)). The cases \(p=1, \;2\) are of particular interest [see e.g. the recent monographs of M. Ledoux, “The concentration of measure phenomenon” (2001; Zbl 0995.60002), and C. Villani, “Topics in optimal transportation” (2003; Zbl 1106.90001), and the references there].

Most of the investigations are based on function analytic tools relating \(T_p(C)\) to e.g., Hamilton-Jacobi equations, Sobolev inequalities and other function inequalities [see e.g. S. G. Bobkov, I. Gentil and M. Ledoux, J. Math. Pure Appl. 80, 669–696 (2001; Zbl 1038.35020), and S. G. Bobkov and F. Götze, J. Funct. Anal. 163, 1–28 (1999; Zbl 0924.46027)].

The investigations in the paper under review are concerned with the opposite direction to establish \(T_p(C)\) without referring to other function inequalities. Having in mind well known estimates for the total variation distance, which is considered as the particular Wasserstein distance for the trivial metric, and furthermore the tail behaviour for Gaussian laws, the following questions are motivated: 1. Under which conditions is finiteness of \(\iint e^{\delta d^2}d\mu\otimes \mu \) necessary and sufficient for \(\mu \in T_1(C)\)? 2. Let \(\mathbb{P}_x^n\) denote the law of a homogeneous Markov chain on \(E^n\), assume the transition kernels \(P(x, \cdot)\) to fulfil \(T_p(C)\) for all \(x\). Under which condition is \(\mathbb{P}_x^n \in T_p(C)\)? 3. In contrast to the afore mentioned investigations, how can \(T_2(C)\) be established if a log-Sobolev inequality is not available? These questions are the main features of this paper. Section 2 contains the theoretical part which is applied in Sections 3 and 4 to random dynamical systems and to diffusions. Section 5, quite independent, is concerned with a direct approach to \(T_2(C)\) for diffusions, using stochastic calculus and martingales, in particular Girsanov’s transformation as main tools.

Most of the investigations are based on function analytic tools relating \(T_p(C)\) to e.g., Hamilton-Jacobi equations, Sobolev inequalities and other function inequalities [see e.g. S. G. Bobkov, I. Gentil and M. Ledoux, J. Math. Pure Appl. 80, 669–696 (2001; Zbl 1038.35020), and S. G. Bobkov and F. Götze, J. Funct. Anal. 163, 1–28 (1999; Zbl 0924.46027)].

The investigations in the paper under review are concerned with the opposite direction to establish \(T_p(C)\) without referring to other function inequalities. Having in mind well known estimates for the total variation distance, which is considered as the particular Wasserstein distance for the trivial metric, and furthermore the tail behaviour for Gaussian laws, the following questions are motivated: 1. Under which conditions is finiteness of \(\iint e^{\delta d^2}d\mu\otimes \mu \) necessary and sufficient for \(\mu \in T_1(C)\)? 2. Let \(\mathbb{P}_x^n\) denote the law of a homogeneous Markov chain on \(E^n\), assume the transition kernels \(P(x, \cdot)\) to fulfil \(T_p(C)\) for all \(x\). Under which condition is \(\mathbb{P}_x^n \in T_p(C)\)? 3. In contrast to the afore mentioned investigations, how can \(T_2(C)\) be established if a log-Sobolev inequality is not available? These questions are the main features of this paper. Section 2 contains the theoretical part which is applied in Sections 3 and 4 to random dynamical systems and to diffusions. Section 5, quite independent, is concerned with a direct approach to \(T_2(C)\) for diffusions, using stochastic calculus and martingales, in particular Girsanov’s transformation as main tools.

Reviewer: Wilfried Hazod (Dortmund)

### MSC:

60E15 | Inequalities; stochastic orderings |

28A35 | Measures and integrals in product spaces |

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |

60G15 | Gaussian processes |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

60J60 | Diffusion processes |

### Keywords:

transportation cost-information inequalities; random dynamical systems; diffusions; Girsanov’s transformations
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\textit{H. Djellout} et al., Ann. Probab. 32, No. 3B, 2702--2732 (2004; Zbl 1061.60011)

### References:

[1] | Bobkov, S., Gentil, I. and Ledoux, M. (2001). Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. (9) 80 669–696. · Zbl 1038.35020 |

[2] | Bobkov, S. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28. · Zbl 0924.46027 |

[3] | Capitaine, M., Hsu, E. P. and Ledoux, M. (1997). Martingale representation and a simple proof of logarithmic Sobolev inequality on path spaces. Electron Comm. Probab. 2 71–81. · Zbl 0890.60045 |

[4] | Dembo, A. (1997). Information inequalities and concentration of measure. Ann. Probab. 25 927–939. · Zbl 0880.60018 |

[5] | Feyel, D. and Ustunel, A. S. (2002). Measure transport on Wiener space and Girsanov theorem. C. R. Acad. Sci. Paris Sér. I Math. 334 1025–1028. · Zbl 1036.60004 |

[6] | Feyel, D. and Ustunel, A. S. (2004). The Monge–Kantorovitch problem and Monge–Ampère equation on Wiener space. Probab. Theory Related Fields . · Zbl 1055.60052 |

[7] | Gentil, I. (2001). Inégalités de Sobolev logarithmiques et hypercontractivité en mécanique statistique et en E.D.P. Thèse de doctorat, Univ. Paul Sabatier Toulouse. |

[8] | Ledoux, M. (2001). The Concentration of Measure Phenomenon . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 |

[9] | Ledoux, M. (2002). Concentration, transportation and functional inequalities. |

[10] | Marton, K. (1996). Bounding \(\overlined\)-distance by information divergence: A method to prove measure concentration. Ann. Probab. 24 857–866. · Zbl 0865.60017 |

[11] | Marton, K. (1997). A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 556–571. · Zbl 0856.60072 |

[12] | Marton, K. (1998). Measure concentration for a class of random processes. Probab. Theory Related Fields 110 427–439. · Zbl 0927.60050 |

[13] | Massart, P. (2003). Concentration inequalities and model selection. In Saint-Flour Summer School . · Zbl 1032.37045 |

[14] | McDiarmid, C. (1989). On the method of bounded differences. Surveys of Combinatorics (J. Siemons, ed.). London Math. Soc. Lecture Notes Ser. 141 148–188. · Zbl 0712.05012 |

[15] | Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400. · Zbl 0985.58019 |

[16] | Rio, E. (2000). Inégalités de Hoeffding pour les fonctions Lipschitziennes de suites dépendantes. C. R. Acad. Sci. Paris Sér. I Math. 330 905–908. · Zbl 0961.60032 |

[17] | Samson, P. M. (2000). Concentration of measure inequalities for Markov chains and \(\phi\)-mixing process. Ann. Probab. 1 416–461. · Zbl 1044.60061 |

[18] | Talagrand, M. (1996). Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 587–600. · Zbl 0859.46030 |

[19] | Villani, C. (2003). Topics in Optimal Transportation . Amer. Math. Soc., Providence, RI. · Zbl 1106.90001 |

[20] | Wang, F. Y. (2002). Transportation cost inequalities on path spaces over Riemannian manifolds. Illinois J. Math. 46 1197–1206. · Zbl 1031.58022 |

[21] | Wu, L. (2000). A deviation inequality for non-reversible Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 36 435–445. · Zbl 0972.60003 |

[22] | Wu, L. (2002). Essential spectral radius for Markov semigroups. I: Discrete time case. Probab. Theory Related Fields 128 255–321. · Zbl 1056.60068 |

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