Lamperti scaling for fractional Gaussian processes with non-stationary increments

TL;DR

This paper extends the Lamperti transform framework to analyze non-stationary Gaussian processes like scaled sub-fractional and bi-fractional Brownian motions, deriving explicit formulas and establishing their stationarity and mixing properties.

Contribution

It introduces explicit covariance formulas, asymptotic behavior, and mixing rates for Lamperti transforms of non-stationary Gaussian processes, and constructs Langevin type integral processes with self-similarity.

Findings

Explicit covariance formulas for transformed processes

Rapid decorrelation of Lamperti images

Single trajectory reconstruction of ensemble quantities

Abstract

The Lamperti transform offers a powerful bridge between self-similar processes and stationary dynamics, making it especially useful for analyzing anomalous diffusion models that lack stationary increments. In this paper we examine the Lamperti transforms of scaled sub-fractional and bi-fractional Brownian motions, deriving explicit covariance formulas, asymptotic behaviour, and precise exponential mixing rates. We also introduce Langevin type integral processes driven by these Gaussian fields, identify their self-similarity exponents, and show that their Lamperti images again form stationary Gaussian processes with rapid decorrelation. Through inverse Lamperti relations and Birkhoff's theorem, we establish rigorous single trajectory reconstruction of ensemble quantities for the original non-stationary processes. The results extend the scope of the scaled Lamperti framework to Gaussian processes with non-stationary increments and richer dependence structures.