A Distributional Identity for the Bivariate Brownian Bridge: A Nontensor Gaussian Field

The bivariate Brownian bridge, a nontensor Gaussian Field, is defined by B(t1,t2)=W(t1,t2)W(1,1)=0=W(t1,t2)-t1t2W(1,1), where t1,t2∈I=[0,1] and W(t1,t2) is a Brownian sheet. We obtain a distributional identity, a consequence of the Karhunen-Loève expansion for the bivariate Brownian bridge by Fredholm integral equation and Laplace transform approach