On an abandoned property, the land may appreciate faster than the improvements depreciate. As long as that pace exceeds the mortgage interest (if any) and property tax (ibid), it's profitable. There are other possible complications too which might swing it, like income tax deductions.

Sometimes seemingly abandoned buildings are used for one-off occasions which generate income: downtown LA is full of "historic" buildings which the owners allow to dilapidate outside of a few areas they can rent out once or twice a year to a film crew.

edit: another thing -- rich people procrastinate too, in fact many of them can afford to procrastinate on matters like this, and they may rather take a loss than confront the anxiety and work of selling or renting out the property