Stochastic flow for SDEs with jumps and irregular drift term

We consider non-degenerate SDEs with a $\beta$-Holder continuous and bounded drift term and driven by a Levy noise $L$ which is of $\alpha$-stable type. If $\alpha \in [1,2)$ and $\beta \in (1 - \frac{\alpha}{2},1)$ we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise $L$. In our previous paper $L$ was assumed to be non-degenerate, $\alpha$-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes of temperated stable processes

On weak uniqueness for some degenerate SDEs by global LpL^p estimates

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the H\"ormander hypoellipticity condition. In the proof we use global $L^p$-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti-Cupini-Lanconelli-Priola (Math. Z. 266 (2010)) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems

Densities for Ornstein-Uhlenbeck processes with jumps

We consider an Ornstein-Uhlenbeck process with values in R^n driven by a L\'evy process (Z_t) taking values in R^d with d possibly smaller than n. The L\'evy noise can have a degenerate or even vanishing Gaussian component. Under a controllability condition and an assumption on the L\'evy measure of (Z_t), we prove that the law of the Ornstein-Uhlenbeck process at any time t>0 has a density on R^n. Moreover, when the L\'evy process is of $\alpha$-stable type, $\alpha \in (0,2)$, we show that such density is a $C^{\infty}$-function

Well-posedness of semilinear stochastic wave equations with H\"{o}lder continuous coefficients

We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an $\alpha$-H\"{o}lder continuous drift coefficient, if $\alpha \in (2/3,1)$. The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations. We also establish regularizing properties of the transition semigroup associated to the stochastic wave equation by using control theoretic results