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

TL;DR

This paper investigates the well-posedness of critical and supercritical stochastic differential equations with singular coefficients, establishing conditions for existence, uniqueness, and pathwise solutions using advanced harmonic analysis techniques.

Contribution

It introduces sharp conditions for well-posedness of SDEs with singular, critical, and supercritical coefficients, extending previous results with new analytical methods.

Findings

Well-posedness of martingale problem under $ ext{α}+eta ext{≥} 1$

Pathwise uniqueness for SDEs with coefficients in Besov spaces

Application of Littlewood-Paley theory to stochastic analysis

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.