  We put forward a new method for obtaining quantitative lower bounds on the
top Lyapunov exponent of stochastic differential equations (SDEs). Our method
combines (i) an (apparently new) identity connecting the top Lyapunov exponent
to a Fisher information-like functional of the stationary density of the Markov
process tracking tangent directions with (ii) a novel, quantitative version of
H\"ormander's hypoelliptic regularity theory in an $L^1$ framework which
estimates this (degenerate) Fisher information from below by a
$W^{s,1}_{\mathrm{loc}}$ Sobolev norm. This method is applicable to a wide
range of systems beyond the reach of currently existing mathematically rigorous
methods. As an initial application, we prove the positivity of the top Lyapunov
exponent for a class of weakly-dissipative, weakly forced SDE; in this paper we
prove that this class includes the Lorenz 96 model in any dimension, provided
the additive stochastic driving is applied to any consecutive pair of modes.
