  Consider the problem of learning the drift coefficient of a $p$-dimensional
stochastic differential equation from a sample path of length $T$. We assume
that the drift is parametrized by a high-dimensional vector, and study the
support recovery problem when both $p$ and $T$ can tend to infinity. In
particular, we prove a general lower bound on the sample-complexity $T$ by
using a characterization of mutual information as a time integral of
conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic
differential equations, the drift coefficient is parametrized by a $p\times p$
matrix which describes which degrees of freedom interact under the dynamics. In
this case, we analyze a $\ell_1$-regularized least squares estimator and prove
an upper bound on $T$ that nearly matches the lower bound on specific classes
of sparse matrices.
