Let [Formula: see text] be a bounded Lipschitz domain of [Formula: see text]. We consider the complement value problem [Formula: see text] Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in our approach.

中文翻译：

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非本地算子的补值问题

令 [Formula: see text] 是 [Formula: see text] 的有界 Lipschitz 域。我们考虑补值问题[公式：见正文] 在温和条件下，我们证明存在唯一的有界连续弱解。此外，我们给出了解决方案的明确概率表示。半狄利克雷形式理论和热核估计在我们的方法中发挥了重要作用。