We consider fractional partial differential equations posed on the full space $\R^d$. Using the well-known Caffarelli-Silvestre extension to $\R^d \times \R^+$ as equivalent definition, we derive existence and uniqueness of weak solutions. We show that solutions to a truncated extension problem on $\R^d \times (0,\YY)$ converge to the solution of the original problem as $\YY \rightarrow \infty$. Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem. These results pave the way for a rigorous analysis of numerical methods for the full space problem, such as FEM-BEM coupling techniques.

翻译：我们考虑在全部空间上 $\ R ⁇ d$ 上提出的部分部分差异方程式。 此外,我们用众所周知的 Caffarelli- Silvestre 扩展为$\ R ⁇ d\ times\ R\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\