  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.
