  The kinetics of a variety of transport-controlled processes can be reduced to
the problem of determining the mean time needed to arrive at a given location
for the first time, the so called mean first passage time (MFPT) problem. The
occurrence of occasional large jumps or intermittent patterns combining various
types of motion are known to outperform the standard random walk with respect
to the MFPT, by reducing oversampling of space. Here we show that a regular but
spatially heterogeneous random walk can significantly and universally enhance
the search in any spatial dimension. In a generic minimal model we consider a
spherically symmetric system comprising two concentric regions with piece-wise
constant diffusivity. The MFPT is analyzed under the constraint of conserved
average dynamics, that is, the spatially averaged diffusivity is kept constant.
Our analytical calculations and extensive numerical simulations demonstrate the
existence of an {\em optimal heterogeneity} minimizing the MFPT to the target.
We prove that the MFPT for a random walk is completely dominated by what we
term direct trajectories towards the target and reveal a remarkable
universality of the spatially heterogeneous search with respect to target size
and system dimensionality. In contrast to intermittent strategies, which are
most profitable in low spatial dimensions, the spatially inhomogeneous search
performs best in higher dimensions. Discussing our results alongside recent
experiments on single particle tracking in living cells we argue that the
observed spatial heterogeneity may be beneficial for cellular signaling
processes.
