Confined kinetics and heterogeneous diffusion driven by fractional Gaussian noise: A path integral approach

TL;DR

This paper develops a path integral framework to analyze heterogeneous, non-Markovian diffusion driven by fractional Gaussian noise with multiplicative effects, revealing how confinement influences probability distribution.

Contribution

It introduces a path-integral approach for fractional Gaussian noise with multiplicative coefficients, unifies different formulations, and derives kinetic equations considering confinement effects.

Findings

Derived a Gaussian propagator using the Lamperti transform.

Revealed the equivalence between Riemann-Liouville fractional Brownian motion and Langevin models.

Showed that confinement induces an effective drift, concentrating probability in low-noise regions.

Abstract

Many complex systems are described by Langevin-type equations in which the noise exhibits long-range correlations and couples to the system in a state-dependent, multiplicative manner, leading to heterogeneous non-Markovian diffusion. Here, we investigate the problem of diffusion driven by fractional Gaussian noise with a general multiplicative coefficient from a path-integral perspective. Using a stationary-phase approximation, we derive a Gaussian propagator expressed in terms of the Lamperti transform of the process. In the additive limit, our results recover the path-integral representation of fractional Brownian motion based on its Riemann-Liouville formulation and establish its equivalence with the Langevin construction. We further analyze the effect of subordinating the process to a killing rate within the Feynman-Kac framework, and develop a general procedure to derive kinetic equations in terms of effective local Hamiltonians. We show that the interplay between multiplicative diffusion and confinement induces an effective drift term, leading to probability accumulation in regions of low noise amplitude.