A Center Controller reported an aircraft cleared for an approach to a non towered airport deviated from the approach course and descended below the prescribed altitudes and flew below the Minimum Vectoring Altitude.

Narrative

I issued Aircraft X the RNAV XX approach at ZZZ. After the pilot was established on the approach; I issued a communications change to contact ZZZ FSS. After scanning the ZZZ area again; I realized the aircraft was in a turn NE bound; no longer established on the approach; heading into a 7000 ft. MIA. The aircraft made a series of maneuvering turns in the 7000 ft. MIA and climbed and then descended between roughly 6000 ft. and 4200 ft. I issued a low altitude alert and a climb to 7000 ft. and my D-side instructed ZZZ FSS to issue a low altitude alert as well. Neither ZZZ FSS or I was able to reach the pilot; and attempted to have an overflying aircraft broadcast on XXX.X; as well as guard frequency to issue the climb to the pilot.No one was able to get a hold of the pilot and the aircraft didn't climb; but eventually the pilot re-established themselves on the approach and made a safe landing. The pilot never contacted ZZZ FSS; but instead finally transmitted on our freq XXX.X to close out their flight plan on the ground at ZZZ. No recommendations. This appeared to be caused by a mixture of pilot navigational error; possible equipment issues; and a lack of familiarization with the area/approach.