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

2022/01/25 by Zhang, Xiaolong, Zhang, Xicheng
#60H10 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2201.10158

Abstract

Let α∈(0,2) and d∈ℕ. Consider the following stochastic differential equation (SDE) driven by α-stable process in ℝ^d: dX_t=b(X_t)dt+σ(X_t-)d L^α_t, X₀=x∈ℝ^d, where b:ℝ^d→ℝ^d and σ:ℝ^d→ℝ^d⊗ℝ^d are locally γ-Hölder continuous with γ∈(0\vee(1-α)⁺,1], L^α_t is a d-dimensional rotationally invariant α-stable process. Under some dissipative and non-degenerate assumptions on b,σ, 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 \citeMZ20 through showing the strong Feller property and the irreducibility of P_t. It is interesting that when α goes to zero, the diffusion coefficient σ can grow faster than drift b. As applications, we put forward a new heavy-tailed sampling scheme.

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