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Stratifications and iterative differential equations are analogues inpositive characteristic of complex linear differential equations. There are fewexplicit examples of stratifications. The main goal of this paper is toconstruct stratifications on projective or affine curves in positivecharacteristic and to determine the possibilities for their differential Galoisgroups. For the related "differential Abhyankar conjecture" we present partialanswers, supplementing the literature. The tools for the construction ofregular singular stratifications and the study of their differential Galoisgroups are $p$-adic methods and rigid analytic methods using Mumford curves andMumford groups. These constructions produce many stratifications anddifferential Galois groups. In particular, some information on the tamefundamental groups of affine curves is obtained.

Stratified Bundles on Curves and Differential Galois Groups in Positive Characteristic