Weak and Strong Well-Posedness of Critical and Supercritical SDEs with Singular Coefficients

Consider the following time-dependent stable-like operator with drift \[\mathscr{L}_{t}\varphi(x) = \int_{\mathbb{R}^{d}}\left[\varphi(x+z)-\varphi(x)-z^{(\alpha)}\cdot\nabla\varphi(x)\right]\sigma(t,x,z)\nu_{\alpha}(dz)+b(t,x)\cdot\nabla\varphi(x),\] where \(d\geq 1\), \(\nu_{\alpha}\) is an \(\alpha\)-stable type Levy measure with \(\alpha\in(0,1]\) and \(z^{(\alpha)}={\bf 1}_{\{\alpha=1\}}{\bf 1}_{\{|z|\leq 1\}}z\), \(\sigma\) 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 \(\mathscr{L}_{t}\) under the sharp balance condition \(\alpha+\beta\geq 1\), where \(\beta\) is the Holder 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.

This talk is based on a joint paper with Longjie Xie of Jiangsu Normal University.

Event Topic

Stochastic & Multiscale Modeling and Computation