Singular SDEs with critical non-local and non-symmetric Lévy type generator

In this work, by using Levi's parametrix method we first construct the fundamental solution of the critical non-local operator perturbed by gradient. Then, we use the obtained estimates to prove the pathwise uniqueness of strong solutions for stochastic differential equation driven by Markov process with irregular coefficients, whose generator is a non-local and non-symmetric Lévy type operator.