Heat kernel of supercritical nonlocal operators with unbounded drifts

— Let α ∈ (0, 2) and d ∈ N. Consider the following stochastic differential equation (SDE) in R^d: dXt = b(t, Xt) dt + a(t, X_t−) dL⁽_t^α), X₀ = x, where L⁽α⁾ is a d-dimensional rotationally invariant α-stable process, b : R₊ × R^d → R^d and a : R₊ × R^d → R^d ⊗ R^d are Hölder continuous functions in space, with respective order β, γ ∈ (0, 1) such that (β ∧ γ) + α > 1, uniformly in t. Here b may be unbounded. When a is bounded and uniformly elliptic, we show that the unique solution Xt(x) of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range α ∈ (0, 1). Our proof is based on ad hoc parametrix expansions and probabilistic techniques.