Ultra-thin magnetic tunnel junctions with low resistive MgO tunnel barriers are prepared to examine their stability under large current stress. The devices show magnetoresistance ratios of up to 110 % and an area resistance product of down to 4.4 ohm micrometer squared. If a large current is applied, a reversible resistance change is observed, which can be attributed to two different processes during stressing and one relaxation process afterwards. Here, we analyze the time dependence of the resistance and use a simple model to explain the observed behavior. The explanation is further supported by numerical fits to the data in order to quantify the timescales of the involved phenomena.