Heterogeneous Planar Diffusion with Axisymmetric Power-Law Decaying Diffusion Coefficient Under Stochastic Resetting

We analyze two-dimensional isotropic heterogeneous diffusion with a radially decaying diffusion coefficient D(ρ)=D0/ρ2, subject to Poissonian stochastic resetting to the origin. Since multiplicative noise makes the process interpretation-dependent, we treat the Itô, Stratonovich, and Hänggi–Klimontovich prescriptions within a single approach. For each, we obtain the exact probability density function, the non-equilibrium stationary state induced by resetting, the mean squared displacement, and the first-passage properties. The non-equilibrium stationary states are governed by modified Bessel functions and depart from the Laplace form found for one-dimensional homogeneous diffusion under resetting, while the mean squared displacements saturate at long times as ⟨ρ2⟩∼r¯−1/2, r¯ being the resetting rate, contrary to the behavior for the one-dimensional homogeneous diffusion, which is of the form ⟨x2⟩∼r¯−1. The non-equilibrium stationary states also depart from those for the one-dimensional heterogeneous diffusion processes, while the corresponding mean squared displacements behave the same. The mean first-passage time to an absorbing boundary exhibits an optimal resetting rate at which it is minimized. In addition, we incorporate memory effects through a subordination (continuous-time random walk) approach with a power-law kernel, which results from the long-tailed waiting time of the particle. Such a two-dimensional process with memory and resetting has not been analyzed before in the literature. We derive the fractional Fokker–Planck equation together with the non-equilibrium stationary states, the mean squared displacements expressed via the three-parameter Mittag–Leffler function, and the first-passage statistics. The heterogeneous diffusion process with memory in all three different interpretations of the multiplicative noise has also not been analyzed in the literature for the one-dimensional case.