Transition probabilities of Lévy‐type processes: Parametrix construction

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.