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

Abstract

Consider the following time-dependent stable-like operator with drift $$ \mathscr{L}_t\varphi(x)=\int_{\mathbb{R}^d}\big[\varphi(x+z)-\varphi(x)-z^{(\alpha)}\cdot\nabla\varphi(x)\big]\sigma(t,x,z)\nu_\alpha(d z)+b(t,x)\cdot\nabla \varphi(x), $$ where $d\geq 1$, $\nu_\alpha$ is an $\alpha$-stable type L\'evy measure with $\alpha\in(0,1]$ and $z^{(\alpha)}=1_{\alpha=1}1_{|z|\leq1}z$, $\sigma$ 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 $\mathscr{L}_t$ under the sharp balance condition $\alpha+\beta\geq1$, where $\beta$ is the H\"older 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 $\mathscr{L}_t$. We prove the pathwise uniqueness of strong solutions for such equations when the coefficients are in certain Besov spaces.