Heterogeneous planar diffusion with axisymmetric power-law decaying diffusion coefficient under stochastic resetting

Abstract

We analyse two-dimensional isotropic heterogeneous diffusion with a radially decaying diffusion coefficient $D(ρ)=D_0/ρ^2$, 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 state is governed by modified Bessel functions and departs from the Laplace form found for homogeneous diffusion under resetting, while the mean squared displacement saturates at long times as $\langleρ^2\rangle\sim\bar{r}^{-1/2}$, $\bar{r}$ being the resetting rate. The mean first-passage time to an absorbing boundary exhibits an optimal resetting rate at which it is minimised. In addition, we incorporate memory effects through a subordination (continuous-time random walk) approach with a power-law kernel, deriving the fractional Fokker--Planck equation together with the non-equilibrium stationary state, the mean squared displacement expressed via the three-parameter Mittag-Leffler function, and the first-passage statistics; the optimal resetting rate increases as the memory exponent decreases.