## Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type.(English)Zbl 1287.60076

The paper deals with a $$d$$-dimensional stochastic differential equation of jump type (jump-diffusion) of the form $dX_t = b_t(X_t)dt+\sigma_t(X_t)dW_t+\int_{\mathbb R^d\backslash\{0\}}f_t(X_{t-},y) \tilde N(dt,dy),$ where $$W$$ is a $$d$$-dimensional standard Wiener process and $$\tilde N$$ is a compensated Poisson random measure. The latter is such that all moments of solutions to this equation exist. The author then studies the Fokker-Planck type equation connected to this jump-diffusion where the interest is in extending existence and uniqueness results to irregular coefficients. Under the assumptions that the drift $$b$$ is in some (appropriate) Sobolev space and that $$\sigma$$ and $$f$$ are Lipschitz continuous, it is proven that a unique generalized stochastic flow exists. This result then allows to obtain the existence and uniqueness of $$L^p$$-valued or measure-valued solutions to the Fokker-Planck equation.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) 60J75 Jump processes (MSC2010)
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