On Davie’s uniqueness for some degenerate SDEs

We consider singular SDEs like \begin{equation} dZ_t = b(t, Z_t) dt + A Z_t dt + \sigma (t) d{L}_t , \;\; t \in [0,T], \;\; Z_0 =x \in \mathbb {R}^n, \end{equation} where $A$ is a real $n \times n$ matrix, i. e., $A \in {\mathbb {R}}^n \otimes {\mathbb {R}}^n$, $b$ is bounded and Hölder continuous, $\sigma \colon [0,\infty ) \to {\mathbb {R}}^n \otimes {\mathbb {R}}^d$ is a locally bounded function and $L= ({L}_t)$ is an $\mathbb {R}^d$-valued Lévy process, $1 \le d \le n$. We show that strong existence and uniqueness together with $L^p$-Lipschitz dependence on the initial condition $x$ imply Davie’s uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved for (0.1) when $n=d$, $A=0$ and $\sigma (t) \equiv I$. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator $\frac {1}{2} \triangle _v f +$ ${v \cdot \partial _{x}f}$ $+F(x,v)\cdot \partial _{v}f$ when $n =2d$ and $L$ is an ${\mathbb {R}}^d$-valued Wiener process. For such equations strong existence and uniqueness are known under Hölder type conditions on $b$. We show that in addition also Davie’s uniqueness holds.