  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, which
in many cases are optimal bounds. In some situation (especially in
one-dimensional SDEs' cases), the discarded measure-zero set can be chosen to
be a measure-zero set of the underlying Brownian motion uniform for all initial
points $X_0=x$.
