  In light of the recent swampland conjectures, we explore quantum cosmology
and eternal inflation beyond the slow roll regime. We consider a model of a
closed universe with a scalar field $\phi$ in the framework of tunneling
approach to quantum cosmology. The scalar field potential is assumed to have a
maximum at $\phi=0$ and can be approximated in its vicinity as $V(\phi)\approx
3H^{2}-\frac{1}{2}m^{2}\phi^{2}$. Using the instanton method, we find that for
$m<2H$ the dominant nucleation channel for the universe is tunneling to a
homogeneous, spherical de Sitter space. For larger values of $m/H$, the most
probable tunneling is to an inhomogeneous closed universe with a domain wall
wrapped around its equator. We determine the quantum state of the field $\phi$
in the nucleated universe by solving the Wheeler-DeWitt equation with tunneling
boundary conditions. Our results agree with earlier work which assumed a
slow-roll regime $m\ll H$. We finally show that spherical universes nucleating
with $m<2H$ undergo stochastic eternal inflation with inflating regions forming
a fractal of dimension $d>2$. For larger values of $m$ the field $\phi$ is
unstable with respect to formation of domain walls and cannot be described by a
perturbative stochastic approach.
