On convergence of 1D Markov diffusions to heavy-tailed invariant density

Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution whose density on the half line has a polynomial decay at infinity. Starting from a standard recipe, which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.