We study the long-time regime of the prediction with expert advice problem in both full information and adversarial bandit feedback setting. We show that with full information, the problem leads to second order parabolic partial differential equations in the Euclidean space. We exhibit solvable cases for this equation and discuss the optimal behavior of both agents. In the adversarial bandit feedback setting, we show that the problem leads to second order parabolic equations in the Wasserstein space which allows us to obtain novel regret bounds.