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 $\mathbf{x}(t)$ of the positions of the particle non Markovian. Using scaling arguments, a Gaussian approximation and numerical methods, we determine the mean first passage time $⟨\mathbf{T}⟩$ of the particle to a target of radius $a$ and show in particular that $⟨\mathbf{T}⟩\sim a^{-1\mathrm{/}2}$ for $a\to 0$, in contrast with the classical logarithmic divergence obtained in the case of an isotropic $2D$ Brownian particle.