Let $H^{*}(BO(n), \mathbf{Z}^{t})$ be the graded cohomology group of the classifying space $BO(n)$ with twisted integer coefficients. Then $H^{*}(BO(n); \mathbf{Z}) \bigoplus H^{*}(BO(n); \mathbf{Z}^{t})$ has a structure of a $\mathbf{Z} \bigoplus \mathbf{Z}_{2}$ graded ring. In the paper this ring is described in terms of generators and relations. It extends the results on the integer cohomology ring $H^{*}(BO(n); \mathbf{Z})$ derived in [B] and [F].

We consider a differential equation on the Wiener space. We obtain some inequalities for capacity under the transformations of the Wiener space induced by the solutions of differential equation and show the existence of local solutions by using the inequalities for capacity.