Will eventually resolve to the party which won the popular vote for the race, as determined by the local, state, and ultimately federal governments.
In the case the race is called by the major news organizations, the market may optionally resolve early.
This market covers events up to and including Election Day, November 5: if Party A's candidate X wins the popular vote that day, but intervening events cause the next inaugurated Senator to be from Party B, the market will resolve to Party A.
If candidate X switches party affiliations, the switch will not affect the outcome if it is after the beginning of Election Day, local time.
Analysis from the Manifold newsletter:
Minnesota: Sen. Amy Klobuchar (D) vs. Royce White (R)
Democrat win chances per models: 98% (The Hill), 98% (538), 97% (Split Ticket)
Expert forecasts: Solid D (Cook), Safe D (Sabato), Solid D (Inside Elections)
Polling average (538): Klobuchar+11
Election result in 2018, a Democratic wave year: D+24 (Klobuchar vs. Newberger)
Election result in 2012, amid Obama's reelection: D+35 (Klobuchar vs. Bills)
Recent Minnesota presidential election results: Biden+7 (2020), Clinton+1.5 (2016), Obama+8 (2012)
Commentary:
In another universe, Minnesota is a competitive state, but people forget that Amy Klobuchar is a really strong candidate in her home state.