arXiv:2611.14692v1 Announce Type: new Abstract: We study (unreflected and reflected) L\'evy processes that are modified at independent Poisson review epochs by a random state-dependent reset map. The first model is unreflected: between resettings the level evolves as a L\'evy process, and at a resetting epoch the new level is $F$ of the level seen just before the resetting. The second model is the reflected L\'evy process at $0$, but with reset maps chosen from a wider transform-solvable class. In this work, we give contraction-type ergodicity criteria for the existence of the steady-state. We identify this steady-state of the system as the solution of certain stochastic fixed-point equation. This allows to find the Laplace-transform, moments recursions, and regularly-varying-tail formulae. For non-reflected case, the examples include affine deterministic resets $F(x)=cx+b$, random-coefficient resets $F(x)=Cx+b$ with two-point $C$ for the finite-activity L\'evy process, spectrally positive L\'evy flight, reflected Brownian process. For the reflected cased we analyse affine resetting mechanism $F(x)=cx$, the alternating resetting $F(x)=(B-x)^+$, and uniform proportional resetting $F(x)=Cx$ with $C$ having uniform distribution.
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