  We study solutions to the free stochastic differential equation $dX_t = dS_t
- \half DV(X_t)dt$, where $V$ is a locally convex polynomial potential in $m$
non-commuting variables. We show that for self-adjoint $V$, the law $\mu_V$ of
a stationary solution is the limit law of a random matrix model, in which an
$m$-tuple of self-adjoint matrices are chosen according to the law $\exp(-N
\textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_\beta$ depends
on complex parameters $\beta_1,...,\beta_k$, then the law $\mu_V$ is analytic
in $\beta$ at least for those $\beta$ for which $V_\beta$ is locally convex. In
particular, this gives information on the region of convergence of the
generating function for planar maps.
  We show that the solution $dX_t$ has nice convergence properties with respect
to the operator norm. This allows us to derive several properties of $C^*$ and
$W^*$ algebras generated by an $m$-tuple with law $\mu_V$. Among them is lack
of projections, exactness, the Haagerup property, and embeddability into the
ultrapower of the hyperfinite II$_1$ factor. We show that the microstates free
entropy $\chi(\tau_V)$ is finite.
  A corollary of these results is the fact that the support of the law of any
self-adjoint polynomial in $X_1,...,X_n$ under the law $\mu_V$ is connected,
vastly generalizing the case of a single random matrix.
