  We analyze the oracle complexity of sampling from polynomially decaying
heavy-tailed target densities based on running the Unadjusted Langevin
Algorithm on certain transformed versions of the target density. The specific
class of closed-form transformation maps that we construct are shown to be
diffeomorphisms, and are particularly suited for developing efficient
diffusion-based samplers. We characterize the precise class of heavy-tailed
densities for which polynomial-order oracle complexities (in dimension and
inverse target accuracy) could be obtained, and provide illustrative examples.
We highlight the relationship between our assumptions and functional
inequalities (super and weak Poincar\'e inequalities) based on non-local
Dirichlet forms defined via fractional Laplacian operators, used to
characterize the heavy-tailed equilibrium densities of certain stable-driven
stochastic differential equations.
