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A General Theory of Dimension-Free Harnack Inequalities

  • Feng-Yu Wang
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

This chapter consists of four sections. In the first section we introduce coupling by change of measure and explain how can one use this new coupling argument to establish dimension-free Harnack inequalities and derivative formulas for stochastic processes. In the second section we explain the main idea in establishing derivative formulas using the Malliavin calculus. In the third section we present some links between Harnack inequalities and gradient inequalities: the Harnack inequality with powers implied by the entropy–gradient inequality and the L 1 gradient–gradient inequality, the log-Harnack inequality implied by the L 2 gradient–gradient inequality, and the Harnack-type inequality equivalent to the L 2 gradient inequality. Finally, some applications of Harnack inequalities are summarized in the last section, which include heat kernel estimates, cost–entropy inequalities, and properties of invariant probability measures. To understand the main idea of this study, Brownian motion with drift on ℝ d is considered in each part as a simple example.

Keywords

Harnack Inequality Markov Operator Markov Semigroup Part Formula Geodesic Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Feng-Yu Wang
    • 1
    • 2
  1. 1.School of Mathematical SciencesBeijing Normal UniversityBeijingChina, People’s Republic
  2. 2.Department of MathematicsSwansea UniversitySwanseaUK

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