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

Abstract

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−)dLtα,X0=x∈Rd, where b: Rd→Rd and σ: Rd→Rd⊗Rd are locally γ-Hölder continuous with γ∈(0∨(1−α)+,1], and Ltα 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.