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 general framework for describing a wide range of natural and engineered systems. We analyze the stochastic differential equation associated with this class of models, in which the driving noise 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 The noise, 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, 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 \emph{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 $R(t)$. From this equation, exact in the well known Poissonian case, both local and global properties of the PDF can be readily inferred. For example, the temporal behavior of the PDF is directly controlled by that of the rate function $R(t)$: if the waiting-time distribution decays as a power law with exponent $\mu>2$, then $R(t)\to const$ and the system converges to the Poissonian equilibrium. By contrast, for $\mu<2$, the rate decays in time and the effective diffusion induced by the noise slowly weakens, without leading to a stationary state. Numerical experiments confirm its remarkable accuracy even far beyond regimes where a naive time-scale separation would justify it.