Tail exponent conjecture for NESS under power-law resetting

Prove that for a one-dimensional Brownian particle subject to the position-dependent resetting rate r(x)=r0|x|^λ, the non-equilibrium steady-state density exhibits asymptotic decay of the form exp(−|x|^{λ/2+1}).

Background

By estimating NESS numerically for several values of λ, the authors observe that the tails fit the form e{−|x|α} with α depending on λ. From these observations, they propose the hypothesis α=λ/2+1 for the tail exponent.

They explicitly state that this hypothesis remains to be proven, framing a clear conjecture about the asymptotic behavior of NESS under power-law resetting that would unify and extend known special cases (λ=0 and λ=2).

References

Following these results, we hypothesize that for a resetting rate r(x)\sim |x|\lambda, the tails of the NESS decay as \sim e{-|x|{\frac{\lambda}{2}+1}. While this hypothesis remains to be proven, it exemplifies the power of our approach, revealing new phenomena and inspiring future work.

— Adaptive Resetting for Informed Search Strategies and the Design of Non-equilibrium Steady-states  (2409.14419 - Keidar et al., 2024) in Section “Prediction and design of non-equilibrium steady states,” paragraph following Fig. 3b