This work explores and develops elements of Stein's method of approximation, in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with $α$-stable ones, $α\in (0,1]$. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincaré-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carré du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincaré-ratio functional of the rotationally invariant $α$-stable laws, $α\in (1,2)$, are obtained; and as such they recover the classical Gaussian setting as $α\to 2$.

66 pages