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

Abstract

Consider the following time-dependent stable-like operator with drift:Ltφ(x)=∫Rd[φ(x+z)−φ(x)−z(α)⋅∇φ(x)]σ(t,x,z)να(dz)+b(t,x)⋅∇φ(x), where d⩾1, να is an α-stable type Lévy measure with α∈(0,1] and z(α)=1α=11|z|⩽1z, σ is a real-valued Borel function on R+×Rd×Rd and b is an Rd-valued Borel function on R+×Rd. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with Lt 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 Lt. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces.