Spectral analysis of a family of symmetric, scale-invariant diffusions with singular coefficients and associated limit theorems

Abstract

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.