Hi, Michele. If this looks like regular "Magic Marker", you can remove it with denatured alcohol from the drugstore. Costs a couple of bucks for a bottle, at most.
You might try washing the area in front of the bathroom door with Murphy's Oil Soap. If that doesn't work, try washing with vinegar in water. If that doesn't work, try washing with ammonia in water. Use about a cup full per gallon of water for each of the last two and rinse with plain water before going from one to the other. One of these should work or at least give you an indication of the nature of the dirt.