Transportation cost-information inequalities and applications to random dynamical systems and diffusions.

*(English)*Zbl 1061.60011Let \((E,d)\) 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)

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