Ergodicity of supercritical SDEs driven by α-stable processes and heavy-tailed sampling

Let alpha is an element of (0,2) and d is an element of N. We consider the stochastic differential equation (SDE) driven by an alpha-stable process dX(t) = b(X-t)dt + sigma(Xt-)dL(t)(alpha), X-0 = x is an element of R-d, where b : R-d -> R-d and sigma : R-d -> R-d circle times Rd are locally gamma-Holder continuous with gamma is an element of (0 boolean OR (1 - alpha)(+), 1], and L-t(alpha) is a d-dimensional symmetric rotationally invariant alpha-stable process. Under certain dissipative and non-degenerate assumptions on b and sigma, we show the V-uniformly exponential ergodicity for the semigroup P-t associated with {X-t (x), t >= 0}. Our proofs are mainly based on the heat kernel estimates recently established in (J. Ec. Polytech. Math. 9 (2022) 537-579) to demonstrate the strong Feller property and irreducibility of P-t. Interestingly, when alpha tends to zero, the diffusion coefficient sigma can increase faster than the drift b. As an application, we put forward a new heavy-tailed sampling scheme.