  A new coupling argument is introduced to establish Driver's integration by
parts formula and shift Harnack inequality. Unlike known coupling methods where
two marginal processes with different starting points are constructed to move
together as soon as possible, for the new-type coupling the two marginal
processes start from the same point but their difference is aimed to reach a
fixed quantity at a given time. Besides the integration by parts formula, the
new coupling method is also efficient to imply the shift Harnack inequality.
Differently from known Harnack inequalities where the values of a reference
function at different points are compared, in the shift Harnack inequality the
reference function, rather than the initial point, is shifted. A number of
applications of the integration by parts and shift Harnack inequality are
presented. The general results are illustrated by some concrete models
including the stochastic Hamiltonian system where the associated diffusion
process can be highly degenerate, delayed SDEs, and semi-linear SPDEs.
