= 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> are
$$
d\mathbf r=v\mathbf n(\theta)dt+\sqrt{2D}\,d\mathbf W,
\qquad d\theta=\omega dt+\sqrt{2D_r}\,dW_r.
$$
Independent translational and rotational noises are the usual interpretation; the supplied autocorrelations alone would not specify arbitrary cross-correlated noises.
The complete state $(\mathbf r,\theta\bmod2\pi)$ 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 <4/d/solution>.
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