  Monolayers of transition metal dichalcogenides (TMDs) have a remarkable
excitonic landscape with deeply bound bright and dark exciton states. Their
properties are strongly affected by lattice distortions that can be created in
a controlled way via strain. Here, we perform a joint theory-experiment study
investigating exciton diffusion in strained tungsten disulfide (WS$_2$)
monolayers. We reveal a non-trivial and non-monotonic influence of strain.
Lattice deformations give rise to different energy shifts for bright and dark
excitons changing the excitonic landscape, the efficiency of intervalley
scattering channels, and the weight of single exciton species to the overall
exciton diffusion. We predict a minimal diffusion coefficient in unstrained
WS$_2$ followed by a steep speed-up by a factor of 3 for tensile biaxial strain
at about 0.6\% strain - in excellent agreement with our experiments. The
obtained microscopic insights on the impact of strain on exciton diffusion are
applicable to a broad class of multi-valley 2D materials.
