Can a police officer obtain a warrant or conduct an investigation for a misdemeanor that occurred six years ago and the statue of limitation has been up for four years now?
Given the abstract and hypothetical nature of the question, no one here can rule out the possibility, but it's unlikely.
Also, you asked about two different things: (1) obtaining a warrant; and (2) conducting an investigation. There is
never any time limit on conducting an investigation. People (including law enforcement officers) conduct investigations of thing, including crimes, that happened decades and centuries ago.
As far as a warrant, you didn't say whether you're concerned with a search warrant or an arrest warrant. If the statute of limitations has truly expired, then I can think of any circumstances that would allow for the issuance of an arrest warrant. A search warrant might be incrementally more likely, but the likelihood would be extremely low.