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

Abstract

Consider the following time-dependent stable-like operator with drift $${\mathcal{L}}_t\phi (x)={\int }_{{\mathbb{R}}^d}[\phi (x+z)-\phi (x)-z^{(\alpha )}\cdot \nabla \phi (x)]\sigma (t,x,z){\nu }_{\alpha }(dz)+b(t,x)\cdot \nabla \phi (x),$$ where $d\ge 1$, ${\nu }_{\alpha }$ is an $\alpha$-stable type Lévy measure with $\alpha \in (0,1]$ and $z^{(\alpha )}=1_{\alpha =1}{1}_{|z|\le 1}z$, $\sigma$ is a real-valued Borel function on ${\mathbb{R}}_{+}\times {\mathbb{R}}^d\times {\mathbb{R}}^d$ and $b$ is an ${\mathbb{R}}^d$-valued Borel function on ${\mathbb{R}}_{+}\times {\mathbb{R}}^d$. By using the Littlewood-Paley theory, we establish the well-posedness for the martingale problem associated with ${\mathcal{L}}_t$ under the sharp balance condition $\alpha +\beta \ge 1$, where $\beta$ 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 ${\mathcal{L}}_t$. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces.