  We study an inverse problem for the fractional wave equation with a potential
by the measurement taking on arbitrary subsets of the exterior in the
space-time domain. We are interested in the issues of uniqueness and stability
estimate in the determination of the potential by the exterior
Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative
unique continuation properties for the fractional Laplacian. For the stability,
we also prove that the log type stability estimate is optimal. The log type
estimate shows the striking difference between the inverse problems for the
fractional and classical wave equations in the stability issue. The results
hold for any spatial dimension $n\in \N$.
