Let \(\varOmega \subset {{\mathbb {R}}}^n\) (\(n\ge 1\)) be a bounded open set with a Lipschitz continuous boundary. In the first part of the paper, using the method of bilinear forms we give a characterization of the realization in \(L^2(\varOmega )\) of the fractional Laplace operator \((-\Delta )^s\) (\(0<s<1\)) with the nonlocal Neumann and Robin exterior conditions. Contrarily to the classical local case \(s=1\), it turns out that the nonlocal (Robin and Neumann) exterior conditions can be incorporated in the form domain. We show that each of the above operators generates a strongly continuous submarkovian semigroup which is also ultracontractive. In the second part, we prove that the semigroup corresponding to the nonlocal Robin exterior condition is always sandwiched between the fractional Dirichlet semigroup and the fractional Neumann semigroup.

令\（\ varOmega \ subset {{\ mathbb {R}}} ^ n \）（\（n \ ge 1 \））是具有Lipschitz连续边界的有界开放集。在本文的第一部分中，我们使用双线性形式的方法对分数拉普拉斯算子\（（-\ Delta）^ s \）（\（L ^ 2（\ varOmega）\）中的实现进行了表征。\（0 <s <1 \））具有非局部Neumann和Robin外部条件。与经典的本地情况\（s = 1 \）相反，事实证明，非局部（Robin和Neumann）外部条件可以并入形式域。我们表明，上述每个算子都会生成一个强连续的亚马尔科夫半群，这也是超收缩的。在第二部分中，我们证明了与非局部Robin外部条件相对应的半群始终夹在分数Dirichlet半群和分数Neumann半群之间。