Mean first-passage time of an anisotropic diffusive searcher

We consider an anisotropic needle-like Brownian particle with nematic symmetry confined in a $2D$ domain. For this system, the coupling of translational and rotational diffusion makes the process ${\bf x} (t)$ of the positions of the particle non Markovian. Using scaling arguments, a Gaussian approximation and numerical methods, we determine the mean first passage time $\langle\mathbf{T}\rangle$ of the particle to a target of radius $a$ and show in particular that $\langle\mathbf{T}\rangle \sim a^{-1/2}$ for $a\to 0$, in contrast with the classical logarithmic divergence obtained in the case of an isotropic $2D$ Brownian particle.