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We consider a microswimmer that moves in two dimensions at a constant speedand changes the direction of its motion due to a torque consisting of aconstant and a fluctuating component. The latter will be modeled by a symmetricL\'evy-stable ($\alpha$-stable) noise. The purpose is to develop a kineticapproach to eliminate the angular component of the dynamics in order to find acoarse grained description in the coordinate space. By defining the jointprobability density function of the position and of the orientation of theparticle through the Fokker-Planck equation, we derive transport equations forthe position-dependent marginal density, the particle's mean velocity and thevelocity's variance. At time scales larger than the relaxation time of thetorque $\tau_{\phi}$ the two higher moments follow the marginal density, andcan be adiabatically eliminated. As a result, a closed equation for themarginal density follows. This equation which gives a coarse-graineddescription of the microswimmer's positions at time scales $t\gg \tau_{\phi}$,is a diffusion equation with a constant diffusion coefficient depending on theproperties of the noise. Hence, the long time dynamics of a microswimmer can bedescribed as a normal, diffusive, Brownian motion with Gaussian increments.

Adiabatic elimination of inertia of the stochastic microswimmer driven by $\alpha-$stable noise