A family of path laws satisfies compact containment when for every finite horizon and every error tolerance there is a compact set which contains the whole path up to that horizon with at least the complementary probability, uniformly over the family. In Euclidean state space this is the displayed condition. It prevents escape to distant regions where local coefficient approximations give no control. A nonnegative Lyapunov function with uniformly bounded generator growth often proves it by the optional stopping theorem.
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