Theory of Probability and Mathematical Statistics
On Davie's uniqueness for some degenerate SDEs
E. Priola
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Abstract: We consider singular SDEs like $dZ_t = b(t, Z_t) dt + A Z_tdt + \sigma(t) d{L}_t , t \in [0,T], Z_0 =x \in \R^n,$ where $A$ is a real $n \times n $ matrix, i.\,e., $A \in {\R}^n \otimes {\R}^n$, $b$ is bounded and Hölder continuous, $\sigma\colon [0,\infty) \to {\R}^n \otimes {\R}^d $ is a locally bounded function and $L= ({L}_t)$ is an $\R^d$-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 \eqref{ss} 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 ${\R}^d$-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.
Keywords: Degenerate stochastic differential equations, path-by-path uniqueness, Hölder continuous drift.
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