You best bet would be to hang gutters beneath it, halfway like this picture
Fastening the gutters will be a little tricky since yours does not have a wall to fasten it.
Probably not a DIY.
http://ornametals.com/
We've been doing copper gutters and zinc gutters in America since 2008 and get very few call backs about replacing stolen gutter pieces. Those who can afford the gutters tend to have adequate home security to protect them.
www.ornametals.com