The induced motion of a probe coupled to a bath with random resettings

We consider a probe linearly coupled to the center of mass of a nonequilibrium bath. We study the induced motion on the probe for a model where a resetting mechanism is added to an overdamped bath dynamics with quadratic potentials. The fact that each bath particle is at random times reset to a fixed position is known for optimizing diffusive search strategies, but here stands for the nonequilibrium aspect of the bath. In the large bath scaling limit the probe is governed by an effective Langevin equation. Depending on the value of the parameters, there appear three regimes: (i) an equilibrium-like regime but with a reduced friction and an increased effective temperature; (ii) a regime where the noise felt by the probe is continuous but non-Gaussian and exhibits fat-tails; (iii) a regime with a non-Gaussian noise exhibiting power-law distributed jumps. The model thus represents an exactly solvable case for the origin of nonequilibrium probe dynamics.