A General Theory of Dimension-Free Harnack Inequalities

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.