  We employ a generalization of Einstein's random walk paradigm for diffusion
to derive a class of multidimensional degenerate nonlinear parabolic equations
in non-divergence form. Specifically, in these equations, the diffusion
coefficient can depend on both the dependent variable and its gradient, and it
vanishes when either one of the latter does. It is known that solution of such
degenerate equations can exhibit finite speed of propagation (so-called
localization property of solutions). We give a proof of this property using a
De Giorgi--Ladyzhenskaya iteration procedure for non-divergence-from equations.
A mapping theorem is then established to a divergence-form version of the
governing equation for the case of one spatial dimension. Numerical results via
a finite-difference scheme are used to illustrate the main mathematical results
for this special case. For completeness, we also provide an explicit
construction of the one-dimensional self-similar solution with finite speed of
propagation function, in the sense of Kompaneets--Zel'dovich--Barenblatt. We
thus show how the finite speed of propagation quantitatively depends on the
model's parameters.
