S i l e n t P o e m
To my knowlege they haven't done a Clown based character.
Somebody missed out on the hilariously awesome Clown-9 storyline in the Spider-Man newspaper strip earlier this year!
The Joker first appeared in
Batman #1 in 1940. Timely Publications, which would later become Marvel, had only been in business about six months at that time and had only a single title (originally
Marvel Comics, changed to
Marvel Mystery Comics in December of '39). There were no clown-based characters that I'm aware of. So the Joker is absolutely a DC character, period.
But!
Marvel Comics #1 features the first appearance of Namor the Sub-Mariner (sort of, he was slated to appear in a giveaway comic earlier that year but the plans fell through and only a few copies were printed, none were distributed). Aquaman didn't show up until 1941, two years later. The two are, uh, similar. So you could argue that Aquaman is inspired by/copied from/a tribute to the Marvel character Namor.
Then there's DC's Swamp Thing and Marvel's more amusingly titled Man-Thing, which both appeared around the same time and may both have been inspired by an obscure character called the Heap.
Marvel's Thor bears an
uncanny resemblance to a version of Thor (the deity, not the Marvel character) found in an old DC adventure comic. Both were drawn by Jack Kirby, but the DC version is older.
Cobra, The Ruthless Terrorist Organization Determined To Rule The World (TM), is a pastiche of Marvel's Hydra and DC's Kobra. Marvel had a hand in the G.I. Joe relaunch back in the 80's as they were working on the tie-in comic (which was originally to have been a young Avengers spin-off featuring Hydra as the villain, they just renamed a bunch of the characters to create Cobra and the Joes).
Short answer here is that the comics industry is full of people jumping between companies, copying ideas or reusing things that didn't work for the competition. No one company is better or worse at it.
Seriously though Joker and Harley (who originated in B:TAS) are totally DC and always have been your girlfriend is crazy brah.