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We study the dynamics of an overdamped Brownian particle in a thermal baththat contains a dilute solution of active particles. The particle moves in aharmonic potential and experiences Poisson shot-noise kicks with specifiedamplitude distribution due to moving active particles in the bath. From theFokker-Planck equation for the particle dynamics we derive the stationarysolution for the displacement distribution along with the momentscharacterizing mean, variance, skewness, and kurtosis, as well as finite timefirst and second moments. We also compute an effective temperature through thefluctuation-dissipation theorem and show that equipartition theorem holds forall zero-mean kick distributions, including those leading to non-Gaussianstationary statistics. For the case of Gaussian-distributed active kicks wefind a re-entrant behaviour from non-Gaussian to Gaussian stationary states anda heavy-tailed leptokurtic distribution across a wide range of parameters asseen in recent experimental studies. Further analysis reveals statisticalsignatures of the irreversible dynamics of the particle displacement in termsof the time asymmetry of cross-correlation functions. Fruits of our work is thedevelopment of a compact inference scheme that may allow experimentalists toextract the rate and moments of underlying shot-noise solely from thestatistics of the particle position.

Brownian particle in a Poisson-shot-noise active bath: exact statistics, effective temperature, and inference