  Constructions of numerous approximate sampling algorithms are based on the
well-known fact that certain Gibbs measures are stationary distributions of
ergodic stochastic differential equations (SDEs) driven by the Brownian motion.
However, for some heavy-tailed distributions it can be shown that the
associated SDE is not exponentially ergodic and that related sampling
algorithms may perform poorly. A natural idea that has recently been explored
in the machine learning literature in this context is to make use of stochastic
processes with heavy tails instead of the Brownian motion. In this paper we
provide a rigorous theoretical framework for studying the problem of
approximating heavy-tailed distributions via ergodic SDEs driven by symmetric
(rotationally invariant) $\alpha$-stable processes.
