Splash potion particles stay on horse when potion is done.

The duration of the splash potion depends on the distance of you and your horse while the potion explodes.
The horse is simply nearer at the potion, thus it's effects are a bit longer lasting.
Look at the wall clock, not on your inventory, if it's a 3:00 speed potion and the horse is still bubbling after 3:00, then it's a bug.

oooo. Ok, thank you. I did not know that.