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

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 $2<sup>{\,n-1}$ 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.