Brownian covariances on an interval: a complete endpoint classification
Abstract
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 2×2 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.
Related
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BibTeX
@misc{raket2026brownian,
title = {{Brownian covariances on an interval: a complete endpoint classification}},
author = {Lars Lau Raket},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36762},
eprint = {2610.36762},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36762},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Brownian covariances on an interval: a complete endpoint classification
AU - Lars Lau Raket
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36762
UR - https://arxiv.org/abs/2610.36762
AB - 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 2×2 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.
ER -