In terms of ownership, a sole proprietorship account is simply an account owned by an individual. Generally, it does not have any attributes that would allow/require you to treat it differently than any other individual account; e.g. you cannot just turn it over to another person based solely on his or her status as a surviving spouse.
With no survivorship provisions, the account belongs to the decedent's estate and would be turned over to the personal representative. He or she may continue to operate the business during the administration of the estate. Any need for the estate to re-register under the assumed name would be a function of state law and might vary from state to state.