  We consider the spatially inhomogeneous Landau equation with soft potentials,
including the case of Coulomb interactions. First, we establish the existence
of solutions for a short time, assuming the initial data is in a fourth-order
Sobolev space and has Gaussian decay in the velocity variable (no decay
assumptions are made in the spatial variable). Next, we show that the evolution
instantaneously spreads mass to every point in its domain. The resulting
pointwise lower bounds have a sub-Gaussian rate of decay, which we show is
optimal. The proof of mass-spreading is based on a stochastic process
associated to the equation, and makes essential use of nonlocality. By
combining this theorem with prior regularity results, we derive two important
applications: $C^\infty$ smoothing in all three variables, even for initial
data with vacuum regions, and a continuation criterion that states the solution
can be extended for as long as the mass and energy densities stay bounded from
above. This is the weakest condition known to prevent blow-up. In particular,
it does not require a lower bound on the mass density or an upper bound on the
entropy density.
