Age-structured hydrodynamics of ensembles of anomalously diffusing particles with renewal resetting

We develop an age-structured hydrodynamic (HD) theory that describes the collective behavior of N≫1 anomalously diffusing particles under stochastic renewal resetting. The theory treats the age of a particle—the time since its last reset—as an explicit dynamical variable and allows for resetting rules that introduce global interparticle correlations. The anomalous diffusion is modeled by the scaled Brownian motion (sBm): a Gaussian process with independent increments, characterized by a power-law time dependence of the diffusion coefficient, D(t)∼t^{2H−1}, where H>0. We apply this theory to three different resetting protocols: independent resetting to the origin (model A), resetting to the origin of the particle farthest from it (model B), and a scaled-diffusion extension of the “Brownian bees” model of Berestycki et al. [Ann. Probab. 50, 2133 (2022)10.1214/22-AOP1578]. In all these models, nonequilibrium steady states are reached at long times, and we determine the steady-state densities. For model A, the (normalized to unity) steady-state density coincides with the steady-state probability density of a single particle undergoing sBM with resetting to the origin. For model B, and for the scaled Brownian bees, the HD steady-state densities are markedly different: In particular, they have compact supports for all H>0. The age-structured HD formalism can be extended to other anomalous diffusion processes with renewal resetting protocols that introduce global interparticle correlations.