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

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.

• The paper introduces a novel path-integral approach to compute transition probabilities for multiplicative, non-Markovian dynamics driven by fractional Gaussian noise.
• It employs a stationary-phase approximation to reduce the propagator to a Gaussian form in the Lamperti variable, confirming classical results under constant diffusion conditions.
• The study derives both local and non-local kinetic equations, elucidating the impact of confinement and state-dependent noise on anomalous diffusion.

Confined Kinetics and Heterogeneous Diffusion Driven by Fractional Gaussian Noise: Path Integral Formalism

Introduction

The study addresses non-Markovian diffusion processes where the system is driven by noise with long-time correlations, specifically fractional Gaussian noise (fGn) parameterized by a Hurst exponent $H$. A central contribution is a general path-integral framework for the calculation of transition probabilities of stochastic dynamics in which the noise couples multiplicatively to the system state. This formalism captures heterogeneous, non-Markovian diffusion scenarios relevant to condensed matter, biological, and financial systems.

Stochastic Models: fBm and Multiplicative Fractional Noise

Fractional Brownian motion (fBm) extends classical Brownian motion by incorporating long-range dependence in increments, with the Hurst exponent controlling anomalous diffusion scaling. The paper revisits the core representations of fBm: the Riemann-Liouville (RL), Mandelbrot-van Ness, and Langevin constructions, each grounded in either integral or differential approaches to fractional calculus. While these frameworks coincide for constant diffusion coefficients, they yield divergent results for time- and state-dependent diffusivities, underscoring the necessity for a unified analytical approach for heterogeneous diffusion.

Multiplicative coupling, $\dot{x}(t) = a(x)\xi_H(t)$, introduces state dependence into the noise amplitude, leading to non-uniform, spatially heterogeneous noise-induced dynamics. This generalization is ubiquitous, underlying processes from population dynamics and finance to anomalous transport in complex materials.

Path Integral Framework and Gaussian Propagator

A central technical development is the derivation of the transition probability (propagator) for multiplicative fGn-driven dynamics using functional integration. The path integral is constructed from the Riemann-Liouville fBm definition, leveraging a stationary-phase (saddle-point) approximation to produce a quadratic action in the auxiliary conjugate noise variable. The propagator is then expressed in terms of the Lamperti transform, which linearizes the multiplicative stochastic differential equation, mapping it to an additive fBm. The resulting transition probability is a Gaussian in the Lamperti variable, with explicit dependence on the initial and final values and the time interval:

$$\mathcal{P}\left(x, t | x_{0}, t_{0}\right) = \frac{1}{a(x)\sqrt{\pi t^{2H}}} \exp\left\{ -\frac{\left(Y[x(t),t]-Y[x_{0},t_{0}] \right)^2}{t^{2H}} \right\}$$

where $Y[x(t),t] = \int \frac{d\tilde{x}}{a(\tilde{x})}$.

For $a(x)$ constant, this expression recovers known propagators for additive fBm and validates the equivalence of path integral and Langevin approaches in that limit.

Local and Non-Local Kinetics

The authors systematically derive the kinetic equation satisfied by the propagator, employing a Feynman-Kac framework. An effective local Hamiltonian, $\mathcal{H}_{\text{eff}}$, is obtained after the stationary-phase reduction, resulting in the local kinetic equation:

$$\frac{\partial}{\partial t}\mathcal{P} = H t^{2H - 1} \partial_x \left[ a(x) \partial_x \left[ a(x) \mathcal{P} \right] \right]$$

This equation captures the time-inhomogeneity and spatial heterogeneity induced by the multiplicative noise coupling. The non-local extension, incorporating the full fractional Riemann-Liouville operator, is shown to correspond to a fundamentally different class of stochastic dynamics akin to generalized continuous-time random walks (CTRWs), with identical MSD scaling $\mathrm{MSD} \sim t^{2H}$ but non-Gaussian statistics in higher moments, highlighting subtleties in kinetic equation construction for non-Markovian processes.

Effects of Confinement and Boundary Conditions

When the process is confined (subject to an infinite killing rate outside a finite domain), the Feynman-Kac formalism is used to impose absorbing boundaries. The normalized propagator within the domain introduces an effective, time-dependent drift term $\mu_H(t)$ in the kinetic equation:

$$\frac{\partial\Psi}{\partial t} - H t^{2H-1} \partial_x\left[a(x) \partial_x\left[a(x)\Psi\right]\right] + \mu_H(t)\Psi = 0$$

This drift arises from the normalization required to preserve probability within the confining domain, leading to a quasi-stationary state and concentration of probability in regions of reduced noise amplitude (lower $\dot{x}(t) = a(x)\xi_H(t)$0). Notably, this is not an equilibrium Boltzmann distribution but emerges solely from temporal normalization dynamics.

Implications and Extensions

This formalism demonstrates that path-integral methodologies, particularly under stationary-phase approximations, allow for unified and exact treatment of multiplicative, non-Markovian stochastic processes even with absorbing boundaries and functional constraints. The reduction to Gaussian statistics in the Lamperti variable and the explicit bridge to the Riemann-Liouville and Langevin perspectives provides a robust analytical foundation for future studies of confined and heterogeneous anomalous transport.

The framework is flexible, extendable to arbitrary, non-Markovian noise structures and multi-dimensional state spaces by changing the cumulant generating functional. This suggests the potential for application in systems with memory, viscoelastic baths, or even interacting particle systems with multiplicative colored noise as well as in the analysis of stochastic thermodynamics with entropy production rates.

Conclusion

The paper establishes a rigorous, general path-integral representation for heterogeneous, non-Markovian diffusion driven by fractional Gaussian noise. The results yield explicit, exact propagators, clarify the distinctions between kinetic equations of differing non-locality, and elucidate the probabilistic consequences of confinement in multiplicative noise systems. The approach bridges several traditional formalisms and supplies a blueprint for future analytical and computational work on complex stochastic models with memory and state-dependent dynamics (2604.10663).