  Let $X$ be a bounded c\`adl\`ag process with positive jumps defined on the
canonical space of continuous paths. We consider the problem of optimal
stopping the process $X$ under a nonlinear expectation operator $\cE$ defined
as the supremum of expectations over a weakly compact family of nondominated
measures. We introduce the corresponding nonlinear Snell envelope. Our main
objective is to extend the Snell envelope characterization to the present
context. Namely, we prove that the nonlinear Snell envelope is an
$\cE-$supermartingale, and an $\cE-$martingale up to its first hitting time of
the obstacle $X$. This result is obtained under an additional uniform
continuity property of $X$. We also extend the result in the context of a
random horizon optimal stopping problem.
  This result is crucial for the newly developed theory of viscosity solutions
of path-dependent PDEs as introduced in Ekren et al., in the semilinear case,
and extended to the fully nonlinear case in the accompanying papers (Ekren,
Touzi, and Zhang, parts I and II).
