 Solution

The term $v\mathbf{n}$ is directed self-propulsion at fixed speed, while the two components of $\mathbf{u}$ generate translational Brownian motion with diffusion coefficient $D$. The angle has systematic angular velocity $\omega$ and rotational diffusion coefficient $D_r$. With independent standard Brownian motions, the precise stochastic differential equations areIndependent translational and rotational noises are the usual interpretation; the supplied autocorrelations alone would not specify arbitrary cross-correlated noises.

The complete state $(\mathbf{r},\theta \mathrm{mod}2\pi )$ is a Markov process: its future increments depend only on its present state and independent future noise. Its generator is $v\mathbf{n}\cdot \nabla +\omega {\partial }_{\theta }+D{\nabla }^2+D_r{\partial }_{\theta }^2$. Position alone generally lacks the Markov property, since an unobserved current orientation carries memory into future displacements. At long scales orientation averaging produces the effective positional diffusion equation in solution.