  In this article we derive Talagrand's $T_2$ inequality on the path space
w.r.t. the maximum norm for various stochastic processes, including solutions
of one-dimensional stochastic differential equations with measurable drifts,
backward stochastic differential equations, and the value process of optimal
stopping problems.
  The proofs do not make use of the Girsanov method, but of pathwise arguments.
These are used to show that all our processes of interest are Lipschitz
transformations of processes which are known to satisfy desired functional
inequalities.
