La. R.S. 9:1964

An interest in income terminates upon the death of the designated beneficiary, or at the expiration of the period of his enjoyment if the interest is for a period less than life. At the termination of an income interest, accumulated or undistributed income that has been or is required to be allocated to the beneficiary shall be paid to the beneficiary or his heirs, legatees, assignees, or legal representatives, except as otherwise provided in this Code.