190.6.a. While a Battlefield is Controlled, its Controller controls its Abilities unless that ability indicates another player does. The controlling player takes responsibility for adding them to the Chain if applicable, and makes all choices required by them unless otherwise specified.
190.6.b. While a Battlefield is Uncontrolled, its Abilities are also Uncontrolled unless that ability indicates a player controls it. If there is no such indication, the Turn Player takes responsibility for adding them to the Chain if applicable, makes all choices required by them unless otherwise specified, and is treated as their Controller if any game rule or effect requires one. Example: The Arena's Greatest is a battlefield that reads "At the start of each player's first Beginning Phase, that player gains 1 point." This ability will usually trigger while the battlefield has no controller. If it does, the Turn Player goes through the steps of adding the ability to the chain and receives priority after doing so, exactly as if they controlled the ability.
190.6.c. If an Ability of a Battlefield indicates that a specific player makes a choice, that player is the Ability's controller. They take responsibility for adding it to the chain if applicable and make all choices required by the ability. They and only they control the ability, regardless of who controls the Battlefield. Example: Abandoned Hall is a battlefield that reads "When a player plays a spell, they may give a unit they control here +1 might this turn." The ability indicates that the player who played the spell makes the choice, so that player takes responsibility for putting the ability on the chain and makes all choices. They control the triggered ability.
190.6.d. "You" in a battlefield's abilities refers to the battlefield's Controller, as does the implied "you" in instructions that don't specify a player like "draw 1." If the battlefield has no Controller, "you" refers to no one, and all such instructions are ignored.