  We introduce a new Neumann problem for the fractional Laplacian arising from
a simple probabilistic consideration, and we discuss the basic properties of
this model. We can consider both elliptic and parabolic equations in any
domain. In addition, we formulate problems with nonhomogeneous Neumann
conditions, and also with mixed Dirichlet and Neumann conditions, all of them
having a clear probabilistic interpretation.
  We prove that solutions to the fractional heat equation with homogeneous
Neumann conditions have the following natural properties: conservation of mass
inside $\Omega$, decreasing energy, and convergence to a constant as $t\to
\infty$. Moreover, for the elliptic case we give the variational formulation of
the problem, and establish existence of solutions.
  We also study the limit properties and the boundary behavior induced by this
nonlocal Neumann condition.
  For concreteness, one may think that our nonlocal analogue of the classical
Neumann condition~$\partial_\nu u=0$ on~$\partial\Omega$ consists in the
nonlocal prescription $$ \int_\Omega \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy=0 \
{\mbox{ for }} x\in\R^n\setminus\overline{\Omega}.$$
