  We consider invariant matrix processes diffusing in non-confining cubic
potentials of the form $V_a(x)= x^3/3 - a x, a\in \mathbb{R}$. We construct the
trajectories of such processes for all time by restarting them whenever an
explosion occurs, from a new (well chosen) initial condition, insuring
continuity of the eigenvectors and of the non exploding eigenvalues. We
characterize the dynamics of the spectrum in the limit of large dimension and
analyze the stationary state of this evolution explicitly. We exhibit a sharp
phase transition for the limiting spectral density $\rho_a$ at a critical value
$a=a^*$. If $a\geq a^*$, then the potential $V_a$ presents a well near
$x=\sqrt{a}$ deep enough to confine all the particles inside, and the spectral
density $\rho_a$ is supported on a compact interval. If $a<a^*$ however, the
steady state is in fact dynamical with a macroscopic stationary flux of
particles flowing across the system. In this regime, the eigenvalues allocate
according to a stationary density profile $\rho_{a}$ with full support in
$\mathbb{R}$, flanked with heavy tails such that $\rho_{a}(x)\sim C_a /x^2$ as
$x\to \pm \infty$. Our method applies to other non-confining potentials and we
further investigate a family of quartic potentials, which were already studied
in Br\'ezin et al. to count planar diagrams.
