We develop a framework for assessing the sensitivity and variability of tsunami inundation characteristics for stochastic physics-based scenarios of mega-thrust subduction earthquakes. The method is applied to the 2011 Tohoku, Japan earthquake, and tested against observed inundation maps at several locations along the Tohoku coast, using 11 different, previously published, rupture models for this devastating tsunamgenic earthquake. The earthquake rupture models differ in fault dimension (length and width), geometry (dip, strike and top-edge depth), as well as asperity characteristics (slip heterogeneity on the fault plane). The resulting source variability allows exploring a wide range of tsunami scenarios for an Mw9 mega-thrust subduction earthquake in the Tohoku region to conduct thorough sensitivity analyses and to quantify the inundation variability. The numerical results indicate a strong influence of the reference source models on inundation variability, and demonstrate significant sensitivity of inundation to the details of the rupture realization. Therefore, relying on a single particular earthquake rupture model as a representative case when varying earthquake source characteristics may lead to under-representation of the variability of potential scenarios. Moreover, the proposed framework facilitates the rigorous development of critical scenarios for tsunami hazard and risk assessments, which are particularly useful for tsunami hazard mapping and disaster preparedness planning.