The Business Analyst (BA) is highly engaged in the initial phases of development. As the subject matter expert, the BA often prioritizes which "slivers" of the project have the most benefit. A "sliver" is a portion of the production ready application that may be released to users. Slivers that bring the greatest returns are targeted first. The BA gathers the requirements and identifies slivers of work using "direct capture of objectives" (DCO) best practices. DCO is a product suite that can be used in any methodology.
Product owners manage a product backlog for a Scrum team. The business analyst (BA) often serves as the product owner, because they have the subject matter expertise required to manage and prioritizes items on the product backlog.
These product backlog items are called "user stories". The product owner is responsible for clarifying the business value, the "return", and how it will be realized based on the prioritization of the user story and its acceptance criteria for completeness. Therefore, the product owner is a stakeholder and has the ability to formally accept or reject user stories submitted as "done" by the team.