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We study the higher regularity of free boundaries in obstacle problems forintegro-differential operators with drift, like $(-\Delta)^s +b\cdot\nabla$, inthe subcritical regime $s>\frac{1}{2}$. Our main result states that once thefree boundary is $C^1$ then it is $C^\infty$, whenever $s\not\in\mathbb{Q}$.  In order to achieve this, we establish a fine boundary expansion forsolutions to linear nonlocal equations with drift in terms of the powers ofdistance function. Quite interestingly, due to the drift term, the powers donot increase by natural numbers and the fact that $s$ is irrational plays alimportant role. Such expansion still allows us to prove a higher order boundaryHarnack inequality, where the regularity holds in the tangential directionsonly.

The fractional obstacle problem with drift: higher regularity of free boundaries