Naixin Liang, S. Peng Oh

Cosmic ray (CR) transport is usually modeled with a single diffusion coefficient, but this description captures only the growth of the variance and not the full transport process. Distinct transport mechanisms can share the same effective diffusion coefficient while producing different particle distributions and approaches to the diffusive limit. This limitation is especially relevant in realistic multiphase, structured, and time-dependent media, and is also reflected in observed environmental variations in CR transport near pulsar wind nebulae, supernova remnants, and molecular clouds. Particle-tracing studies also show clear departures from standard diffusion, including both superdiffusion and subdiffusion. We therefore develop a propagator-based framework centered on $P(x,t)$, the probability distribution of particle positions, or equivalently its Fourier-Laplace transform $P(k,s)$. This object is compact and statistically complete, and naturally exposes memory: the CR flux can depend on earlier gradients when unresolved trapping or phase changes are coarse-grained away. Using the Montroll-Weiss formalism, we show how to measure $P(k,s)$ directly from trajectories, how to recover the associated memory kernel, and how to represent broad kernels efficiently with a Prony expansion. Applied to a multiphase medium, the framework shows that slow regions can regulate escape without dominating the total residence-time budget. We also introduce an accelerated Monte Carlo method for coarse-grained transport, and show that if trapping structures evolve while particles are still sampling them, the static long-time limit need not be reached. This paper provides the foundation for future observational applications, particle-tracing measurements, and CR-MHD closures.