  In this paper we are interested in integro-differential elliptic and
parabolic equations involving nonlocal operators with order less than one, and
a gradient term whose coercivity growth makes it the leading term in the
equation. We obtain Lipschitz regularity results for the associated stationary
Dirichlet problem in the case when the nonlocality of the operator is confined
to the domain, feature which is known in the literature as censored
nonlocality. As an application of this result, we obtain strong comparison
principles which allow us to prove the well-posedness of both the stationary
and evolution problems, and steady/ergodic large time behavior for the
associated evolution problem.
