We study the long-time regime of the prediction 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 Euclidian space. We exhibit solvable cases for this equation. In the adversarial bandit feedback setting, we show that the problem leads to a second order equation in the Wasserstein space. Based on joint works with Ibrahim Ekren and Xin Zhang.