Karhunen–Loève expansion for additive Brownian motions

For Gaussian random fields defined as additive processes based on standard Brownian motions and Brownian bridges, we find their Karhunen–Loève expansions and make connections with related mean centered processes in distribution. Moreover, Pythagorean type distribution identities are established for additive Brownian motions and Brownian bridges. As applications, the corresponding Laplace transform and small deviation estimates are given.