Localization Delocalization Transition in Diffusion with Adaptive Resetting

Stochastic resetting can localize diffusion and generate nonequilibrium steady states, but the conditions under which spatially dependent resetting produces localization remain unclear. Here, we establish a general classification for diffusion under adaptive resetting, where the resetting rate $r(x)$ depends on position. For rates with the asymptotic scaling $r(x)\sim |x|^λ$, we identify a sharp threshold at $λ=-2$. For $λ>-2$, the steady state is localized and exhibits stretched-exponential tails, whereas for $λ<-2$, resetting is asymptotically too weak to localize the particle. Precisely at the marginal scaling $r(x)\sim |x|^{-2}$, a qualitatively new regime emerges: the steady state develops power-law tails with a temperature-dependent exponent and exhibits a finite-temperature delocalization transition. Thus, inverse-square resetting plays the role of the logarithmic potential in equilibrium, establishing a nonequilibrium counterpart of the temperature-driven delocalization seen there.