  We study the almost sure behavior of solutions of stochastic differential
equations (SDEs) as time goes to zero. Our main general result establishes a
functional law of the iterated logarithm (LIL) that applies in the setting of
SDEs with degenerate noise satisfying the weak Hormander condition but not the
strong Hormander condition}. That is, SDEs in which the drift terms must be
used in order to conclude hypoellipticity. As a corollary of this result, we
obtain the almost sure behavior as time goes to zero of a given direction in
the equation, even if noise is not present explicitly in that direction. The
techniques used to prove the main results are based on large deviations applied
to a non-trivial rescaling of the original system. In concrete examples, we
show how to find the proper rescaling to obtain the functional LIL.
Furthermore, we apply the main results to the problem of identifying regular
points for hypoelliptic diffusions. Consequently, we obtain a control-theoretic
criteria for a given point to be regular for the process.
