Stochastic Path Integral for the Active Brownian Particle in a Harmonic Potential

In this work we develop and apply a path integral formulation for the microscopic degrees of freedom obeying stochastic differential equations to an active Brownian particle (ABP) trapped in a harmonic potential. The formalism allows to derive exact analytic expressions for the time-dependent moments, like the mean position and the mean square displacement, including full dependence on initial conditions. In addition, the probability distribution of the particle's position can be evaluated systematically as a series expansion in the propulsion speed. Compared to previous methods relying on eigenfunction expansions of the equivalent Fokker-Planck equation, our method is easier to generalize to more complex situations: it does not rely on eigenfunctions but on a reference state that can be solved analytically, which in our case is the passive Brownian particle in a harmonic potential. We exemplify this versatility by also briefly treating an ABP with an active torque (Brownian circle swimmer, BCS) in a harmonic potential.