Lamperti scaling for fractional Gaussian processes with non-stationary increments

TL;DR

The paper extends the Lamperti transform to scaled self-similar Gaussian processes with non-stationary increments, notably $s$-fBm and $bi$-fBm, and analyzes associated Langevin-type integrals. By introducing a scaling exponent $\alpha$, the scaled Lamperti images become stationary Gaussian fields with explicit covariances that decay exponentially, yielding full ergodicity and strong mixing with rates determined by $\alpha H$ (sub-fractional) or $\alpha \min\{2H-HK,1-HK\}$ (bi-fractional). The authors also derive reconstruction formulas via inverse Lamperti relations, showing single-trajectory data suffices to recover ensemble properties of the original non-stationary processes. Numerical simulations validate stationarity, mixing, and Gaussianity, illustrating the practical utility for anomalous diffusion models with non-stationary increments.

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.

Figures (4)

• Figure 1: Left: sample trajectory of the scaled $s-fBm$ process $S_H(t^\alpha)$ with $H=0.7$, $\alpha=3/2$. Right: trajectory of its Lamperti transform $S_H^{\mathrm{LT}}(t)=e^{-\alpha H t}S_H(e^{\alpha t})$.
• Figure 2: Left: trajectory of the scaled $bi-fBm$ process $B_{H,K}(t^\alpha)$ with $H=0.7$, $K=0.6$, $\alpha=3/2$. Right: trajectory of the Lamperti transform $B_{H,K}^{\mathrm{LT}}(t)=e^{-\alpha HK t}B_{H,K}(e^{\alpha t})$.
• Figure 3: Lamperti $s-fBm$ with $H=0.6$, $\alpha=3$. Left: empirical long time second moment approaching its theoretical limit $2 - 2^{2H-1}$. Right: empirical characteristic function compared with the Gaussian characteristic function.
• Figure 4: Lamperti $bi-fBm$ with $H=0.8$, $K=0.6$, $\alpha=1.5$. Empirical characteristic function compared with the standard Gaussian characteristic function.

Theorems & Definitions (56)

• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5
• proof
• Theorem 2.6
• proof
• Lemma 2.7
• proof