Weak and strong well-posedness of critical and supercritical SDEs with singular coefficients

Consider the following time-dependent stable-like operator with drift: L_tφ(x)=∫R^d[φ(x+z)−φ(x)−z^(α)⋅∇φ(x)]σ(t,x,z)ν_α(dz)+b(t,x)⋅∇φ(x), where d⩾1, ν_α is an α-stable type Lévy measure with α∈(0,1] and z^(α)=1_α=11_|z|⩽1z, σ is a real-valued Borel function on R₊×R^d×R^d and b is an R^d-valued Borel function on R₊×R^d. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with L_t under the sharp balance condition α+β⩾1, where β is the Hölder index of b with respect to x. Moreover, we also study a class of stochastic differential equations driven by Markov processes with generators of the form L_t. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces.