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In this work, by using Levi's parametrix method we first construct thefundamental solution of the critical non-local operator perturbed by gradient.Then, we use the obtained estimates to prove the existence and uniqueness ofstrong solutions for stochastic differential equation driven by Markov processwith irregular coefficients, whose generator is a non-local and non-symmetricL\'evy type operator.

Singular SDEs with critical non-local and non-symmetric L\'evy type generator