Memory and motion: a simple identity in stochastic transport

Since Robert Brown’s observation of the erratic motion of pollen grains, Brownian dynamics have provided a cornerstone for understanding transport at small scales. Theoretical developments by Smoluchowski and Einstein established diffusion as an emergent statistical consequence of stochastic microscopic motion, linking random fluctuations to macroscopic transport coefficients. Within this framework, two complementary descriptors play a central role: temporal correlations, which quantify persistence or memory, and cumulative measures of motion, which quantify the extent of transport over time. These descriptors are traditionally treated as distinct observables, connected indirectly through constitutive modelling or fluctuation–dissipation relations. Here, I show that for a broad class of bounded stochastic systems they are, however, fundamentally constrained. When suitably normalised, correlation and cumulative transition are not independent but reflect complementary aspects of the same underlying dynamics. This relationship follows directly from probability normalisation and causality, and is independent of the detailed form of the stochastic evolution. Framed in this way, the result provides a compact statement of the interplay between persistence and motion, with implications for diffusion, rheology, microrheology, and active-matter transport.