Brownian covariances on an interval: a complete endpoint classification

Brownian motion and the Brownian bridge have covariance kernels that are Green's functions of the negative second derivative on the unit interval, with mixed Dirichlet-Neumann and Dirichlet boundary conditions, respectively. We characterize the Green's functions of all strictly positive real self-adjoint realizations of this operator. The kernels are in one-to-one correspondence with positive-semidefinite $2\times2$ endpoint covariance matrices. We call the associated centered Gaussian processes Brownian processes. Each is a standard Brownian bridge plus the affine interpolation of an independent Gaussian endpoint vector. Singular endpoint covariance matrices give hard endpoint constraints, such as the pinned start $X(0)=0$ of Brownian motion. This parametrization gives explicit operator domains and energy spaces. On every mesh it gives an exact sparse precision matrix, so the likelihood of $n$ observations can be evaluated in $O(n)$ operations. Nonnegative realizations with a null space are treated separately through orthogonally normalized pseudoinverses, which are proper Gaussian covariances.