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

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 associated with this class of models, in which the driving noise (Formula presented) (Formula presented) consists of spike (shot) events, and we derive two exact analytical results. First, we obtain a closed-form expression for the (Formula presented) (Formula presented) -time correlation functions of (Formula presented) (Formula presented), expressed as a sum over all (Formula presented) (Formula presented) ordered partitions of the observation times (proposition 2). Second, using the (Formula presented) (Formula presented) -cumulant formalism, we derive an exact non-local master equation (ME) for the probability density function of the CTRW variable (Formula presented) (Formula presented), valid without invoking diffusive limits, fractional scaling assumptions, or closure hypotheses (proposition 6). 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 universal local ME: 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 (Formula presented) (Formula presented). This formulation naturally accounts for the intrinsic non-Markovian character of renewal processes: different preparation protocols, including aging situations where observation starts between renewal events, are fully captured through the corresponding time-dependent renewal rate. 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 (Formula presented) (Formula presented) : if the waiting-time distribution decays as a power law with exponent (Formula presented) (Formula presented), then (Formula presented) (Formula presented) and the system converges to the Poissonian equilibrium. By contrast, for (Formula presented) (Formula presented), 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.