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

Let α ∈ (0,2) and d ∈ N. We consider the stochastic differential equation (SDE) driven by an α-stable process dXt = b(Xt)dt + σ(Xt−)dLα t, X0 = x ∈ Rd, where b: Rd →Rd and σ: Rd →Rd ⊗ Rd are locally γ-Hölder continuous with γ ∈ (0 ∨ (1 − α)+,1], and Lα t is a d-dimensional symmetric rotationally invariant α-stable process. Under certain dissipative and non-degenerate assumptions on b and σ, we show the V-uniformly exponential ergodicity for the semigroup Pt associated with {Xt (x), t ≥ 0}. Our proofs are mainly based on the heat kernel estimates recently established in (J. Éc. Polytech. Math. 9 (2022) 537–579) to demonstrate the strong Feller property and irreducibility of Pt. Interestingly, when α tends to zero, the diffusion coefficient σ can increase faster than the drift b. As an application, we put forward a new heavy-tailed sampling scheme.