The effective diffusion constant of stochastic processes with spatially periodic noise

2024/07/15 by Stefano Giordano, Giordano, Stefano, Ralf Blossey +1 · 3 citations
Economics, Econometrics and Finance · #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2407.10813

Abstract

We discuss the effective diffusion constant D_{\it eff} for stochastic processes with spatially-dependent noise. Starting from a stochastic process given by a Langevin equation, different drift-diffusion equations can be derived depending on the choice of the discretization rule 0 ≤ α≤ 1. We initially study the case of periodic heterogeneous diffusion without drift and we determine a general result for the effective diffusion coefficient D_{\it eff}, which is valid for any value of α. We study the case of periodic sinusoidal diffusion in detail and we find a relationship with Legendre functions. Then, we derive D_{\it eff} for general α in the case of diffusion with periodic spatial noise and in the presence of a drift term, generalizing the Lifson-Jackson theorem. Our results are illustrated by analytical and numerical calculations on generic periodic choices for drift and diffusion terms.

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• Boltzmann to Lindblad: Classical and Quantum Approaches to Out-of-Equilibrium Statistical Mechanics
• Universal asymptotic solution of the Fokker-Planck equation with time-dependent periodic potentials
• Effective diffusion of Brownian motion in spatially quasi-periodic noise

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