Compact containment
= Compact containment
{title2=$\lim_{R\to\infty}\sup_n\mathbb P(\sup_{t\le T}|X_t^n|>R)=0$}
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>.