Investigation ... who does it?
I can't answer that, but if the question was "Who pays for it?" (the search, at least), the answer would seem to be Australia.
Why is totally beyond me. Both China (with by far the largest number of citizens among the missing) and Malaysia (the operator of the missing aircraft) have been studiously slow in putting up any money towards the search, while Australia, if only because the aircraft MAY have flown into its SAR area of responsibility before crashing, continues to foot the very large bill for the search.
And in the grand tradition of 'you couldn't make this up', Australia has to borrow the money to pay for the search from.... China!