Exact and limit results for the CTRW in presence of drift and position dependent noise intensity

2026-03-16

AI Summary

This study investigates the statistical behavior of continuous-time random walks (CTRWs) with drift and position-dependent noise intensity under non-diffusive, non-stationary conditions. By analyzing the associated stochastic differential equation and employing G-cumulants, ordered partition summation, and the interaction picture formalism, the authors rigorously derive, for the first time, closed-form expressions for arbitrary-order multi-time correlation functions of the driving noise and an exact nonlocal master equation that does not rely on diffusion or fractional scaling assumptions. They further demonstrate that, in the long-time limit, this master equation universally reduces to a local form dependent solely on the instantaneous renewal rate, extending its validity well beyond conventional timescale separation approximations. Numerical simulations confirm the high accuracy of this approximation across a broad parameter regime and recover the classical Poisson master equation in the constant renewal rate limit.

Technology Category

Planning, Routing, and Scheduling: Mixed Discrete/Continuous PlanningSearch and Optimization: Mixed Discrete/Continuous SearchReasoning under Uncertainty: Stochastic Optimization

Application Category

Web Mining and Content Analysis: Content-based information diffusionGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSecurity and Privacy: Large-scale security measurements

Abstract

Continuous-time random walks (CTRWs) with drift and position-dependent jumps provide a highly general framework for describing a wide range of natural and engineered systems. We analyze the stochastic differential equation (SDE) associated with this class of models, in which the driving noise $ξ(t)$ consists of spike (shot) events, and we derive two exact analytical results. First, we obtain a closed-form expression for the $n$-time correlation functions of $ξ(t)$, expressed as a sum over all $2^{\,n-1}$ ordered partitions of the observation times (Proposition~2). Second, using the $G$-cumulant formalism, we derive an \emph{exact} non-local master equation (ME) for the probability density function of the CTRW variable $x(t)$, valid without invoking diffusive limits, fractional scaling assumptions, or closure hypotheses (Proposition~3). In interaction representation, this ME retains the same structural form as that of the standard CTRW without drift or position-dependent jumps. Our main result is the emergence of a \textbf{universal local master equation}: at long times, the exact non-local ME is universally and accurately approximated by a time-local ME whose only coefficient is the instantaneous renewal rate $R(t)$. This approximation reproduces the exact Poissonian ME when $R$ is constant, and numerical experiments confirm its remarkable accuracy even far beyond regimes where a naive time-scale separation would justify it.

Problem

• continuous-time random walk
• position-dependent noise
• drift
• master equation
• non-local dynamics

Innovation

• continuous-time random walk
• position-dependent noise
• exact master equation
• G-cumulant formalism
• universal local approximation