  We discuss a family of time-reversible, scale-invariant diffusions with
singular coefficients. In analogy with the standard Gaussian theory, a
corresponding family of generalized characteristic functions provides a useful
tool for proving limit theorems resulting in non-Gaussian, scale-invariant
diffusions. We apply the generalized characteristic functions in combination
with a martingale construction to prove a simple invariance principle starting
from a spatially inhomogeneous nearest-neighbor random walk.
