  We establish quantitative stability for the nonlocal Serrin overdetermined
problem, via the method of the moving planes. Interestingly, our stability
estimate is even better than those obtained so far in the classical setting
(i.e., for the classical Laplacian) via the method of the moving planes.
  A crucial ingredient is the construction of a new antisymmetric barrier,
which allows a unified treatment of the moving planes method. This strategy
allows us to establish a new general quantitative nonlocal maximum principle
for antisymmetric functions, leading to new quantitative nonlocal versions of
both the Hopf lemma and the Serrin corner point lemma.
  All these tools -- i.e., the new antisymmetric barrier, the general
quantitative nonlocal maximum principle, and the quantitative nonlocal versions
of both the Hopf lemma and the Serrin corner point lemma -- are of independent
interest.
