Daniel W. Stroock

Itô’s Approach

To address the problem of convergence raised at the end of Chapter 1, K. Ito used a technique known as coupling . Given a pair of Borel probability measures \(\mu _1\) and \(\mu _2\) on some metric space \((E,\rho )\), a coupling of \(\mu _1\) to \(\mu _2\) is a pair of E-valued random variables \(X_1\) on \(X_2\) on some probability space \((\varOmega ,\mathcal {B},{\mathbb {P}})\) such that \(\mu _1\) is the distribution of \(X_1\) and \(\mu _2\) is the distribution of \(X_2\). Given such a coupling, one can compare \(\mu _1\) to \(\mu _2\) by looking at

Markov Processes from K. Ito's Perspective (AM-155)

{\mathbb {E}}^{\mathbb {P}}\bigl [\rho \bigl (X_1,X_2\bigr )^p\bigr ]^{\frac{1}{p}}.