Exact moments for a run and tumble particle in a harmonic trap with a finite tumble time

We study the problem of a run and tumble particle in a harmonic trap, with a finite run and tumble time, by a direct integration of the equation of motion. An exact 1D steady state distribution, diagram laws and a programmable Volterra difference equation are derived to calculate any order of moments in any other dimension, both for steady state as well as the Laplace transform in time for the intermediate states. We also use the moments to infer the distribution by considering a Gaussian quadrature for the corresponding measure, and from the scaling law of high order moments.