New characterizations of Brownian motion

In the paper new characterizations of Brownian motion are proved. They generalize and supplement the famous Levi theorem on the characterization of the process of Brownian motion in the class of square integrable continuous martingales. The first characterization (Theorem 1) generalizes the Levi theorem. Two other characterizations (Theorems 2 and 3) are analogues of the Levi theorem, in which the continuity condition is replaced by other conditions.