  We prove existence, uniqueness and optimal regularity of solutions to the
stationary obstacle problem defined by the fractional Laplacian operator with
drift, in the subcritical regime. We localize our problem by considering a
suitable extension operator introduced by L. Caffarelli and L. Silvestre. The
structure of the extension equation is different from the one considered by L.
Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem
for the fractional Laplacian without drift, in that the obstacle function has
less regularity, and exhibits some singularities. To take into account the new
features of the problem, we prove a new Almgren-type monotonicity formula,
which we then use to establish the optimal regularity of solutions.
