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In this note we consider autonomous SDEs admitting smooth invariant measures.We present a method in finding (almost everywhere) good bounds for $\sup\{\|X_t\|: t \in [0, T]\}$ for strong solutions $X_{\cdot}$ to such SDEs, whichin many cases are optimal bounds. In some situation (especially inone-dimensional SDEs' cases), the discarded measure-zero set can be chosen tobe a measure-zero set of the underlying Brownian motion uniform for all initialpoints $X_0=x$.

Bounding the Solutions to Some SDEs via Ergodic Theory