Malliavin introduced his analysis of functionals on the Wiener space to obtain a probabilistic proof of the Hoermander theorem.
Hoermander theorem states that the generalized Heat equation on R^n has a smooth solution whenever the driving function is smooth, provided that the Lie algebra of the heat operator generators span the whole of R^n at each point. Of course, analysis on Wiener spaces have many applications in other areas also. This is a reference to a review of "Stochstic analysis" by Malliavin