Resetting of underdamped Brownian motion

We study underdamped Brownian motion under Poissonian resetting, where we apply two different full resetting procedures. The first one resets both position and velocity to zero, while the second one resets the displacement to zero but takes the velocity after resetting to follow a Maxwell distribution. In both cases, the marginal probability density function (PDF) of the displacement in the stationary state shows exponential tails for large displacements, but the singularities at the mode of the PDF are very different and specific for the corresponding resetting prescription: in the first case, one has p(x)∼|x|^{-1/3}, while in the second one, one gets p(x)∼-ln|x|. The convergence of the PDF to a stationary one is investigated by examining its excess kurtosis as a function of time. Applying two sets of initial conditions for the second resetting procedure, we show that the behavior of the excess kurtosis for short times is highly sensitive to these.