Closed-form NESS for diffusion with power-law position-dependent resetting

Derive closed-form expressions for the non-equilibrium steady-state of a one-dimensional Brownian particle with position-dependent resetting rate r(x)=r0|x|^λ for exponents λ ≠ 0,2, extending the known exact solutions for the constant-rate case (λ=0) and parabolic case (λ=2).

Background

The authors examine non-equilibrium steady states (NESS) arising from diffusion with position-dependent resetting rates, focusing on power-law forms r(x)=r0|x|^λ. Exact NESS are known for constant resetting (λ=0, Laplace distribution) and for parabolic resetting (λ=2).

For general λ, the authors emphasize that closed-form solutions are currently unavailable, though they can estimate NESS numerically from reset-free trajectories using their reweighting approach. Providing closed-form solutions would advance theoretical understanding and enable analytical characterization of NESS beyond special cases.