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

TL;DR

This paper introduces a family of scale-invariant diffusions with singular coefficients, utilizing generalized characteristic functions and martingale methods to establish limit theorems and invariance principles for inhomogeneous random walks.

Contribution

It develops a novel approach using generalized characteristic functions to analyze non-Gaussian, scale-invariant diffusions with singular coefficients, extending classical Gaussian theory.

Findings

Proves limit theorems for a family of non-Gaussian diffusions

Establishes an invariance principle for inhomogeneous random walks

Demonstrates the effectiveness of generalized characteristic functions in this context

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.