  We address the problem of random search for a target in an environment with
space-dependent diffusion coefficient $D(x)$. From a general form of the
diffusion differential operator that includes It\^o, Stratonovich, and
H\"anggi-Klimontovich interpretations of the associated stochastic process, we
obtain the first-passage time distribution and the search efficiency
$\mathcal{E}=\langle 1/t \rangle$. For the paradigmatic power-law diffusion
coefficient $D(x) = D_0|x|^{\alpha}$, with $\alpha<2$, which controls whether
the mobility increases or decreases with the distance from a target at the
origin, we show the impact of the different interpretations. For the
Stratonovich framework, we obtain a closed expression of the search efficiency,
valid for arbitrary diffusion coefficient $D(x)$. We show that a heterogeneous
diffusivity profile leads to lower efficiency than the homogeneous average
level, and the efficiency depends only on the distribution of diffusivity
values and not on its spatial organization, features that breakdown under other
interpretations.
