Stochastic resetting, a diffusive process whose amplitude is reset to the origin at random times, is a vividly studied strategy to optimize encounter dynamics, e.g., in chemical reactions. We here generalize the resetting step by introducing a random resetting amplitude, such that the diffusing particle may be only partially reset towards the trajectory origin, or even overshoot the origin in a resetting step. We introduce different scenarios for the random-amplitude stochastic resetting process and discuss the resulting dynamics. Direct applications are geophysical layering (stratigraphy) as well as population dynamics or financial markets, as well as generic search processes.
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