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We study stochastic differential equations with jumps with no diffusion part.We provide some basic stochastic characterizations of solutions of thecorresponding non-local partial differential equations and prove the Harnackinequality for a class of these operators. We also establish key connectionsbetween the recurrence properties of these jump processes and the non-localpartial differential operator. One of the key results is the regularity ofsolutions of the Dirichlet problem for a class of operators with locally weaklyH\"older continuous kernels.

On a Class of Stochastic Differential Equations With Jumps and Its Properties