We will introduce coupling by change of measures to establish Harnack inequalities and derivative formulas of heat semigroups for stochastic differential equations. In particular, equivalent inequalities are derived for curvature and second fundamental lower bounds on Riemannian manifolds with boundary.
Links:
[1] http://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/de/node/3444
[3] http://www.mpim-bonn.mpg.de/de/node/7138