On weak solution of SDE driven by inhomogeneous singular Lévy noise

We study a time-inhomogeneous SDE inRd\mathbb {R}^ddriven by cylindrical Lévy process with independent coordinates which may have different scaling properties. Such structure of the driving noise makes it strongly spatially inhomogeneous and complicates analysis model significantly. prove that weak solution to is uniquely defined, Markov, has strong Feller property. The heat kernel presented as combination an explicit ‘principal part’ ‘residual part’, subject certainL∞(dx)⊗L1(dy)L^\infty (dx)\otimes L^1(dy)andL-estimates showing this part negligible in short time, sense. main tool construction analytic parametrix method, specially adapted Lévy-type generators spatial inhomogeneities.