A Bestiary of Brownian Processes: A Field Guide to Boundary Conditions and Conditioning
Abstract
The covariance of Brownian motion is the Green kernel of the one-dimensional Laplacian with a pinned left endpoint and a free right endpoint. This correspondence between Green's functions for the Laplacian on an interval and Brownian covariance functions turns out to be a general phenomenon. We use this correspondence to organize Brownian-derived Gaussian processes by three operations: choosing a self-adjoint realization, recentering a path, and conditioning a fixed law. Local Robin and selected nonlocal boundary conditions yield a catalogue of twelve processes with closed-form kernels, Brownian-path constructions, spectra and traces. The periodic and Neumann-Neumann realizations annihilate the constants; projecting the source and imposing zero time average selects the corresponding covariance pseudoinverses. Recentering and conditioning can both give a Brownian path zero time average, but the resulting laws differ: a constraint alone does not specify a process. A rank-one conditioning identity gives a scale-invariant criterion for the covariance effect of a shrinking family of observations. For Brownian finite differences of any fixed positive order, the covariance effect vanishes uniformly as the spacing tends to zero. For m-fold integrated Brownian motion, derivative constraints of orders 1,…,m remain detectable at interior times. For simulation, we identify which grid constructions are exact and which are approximate.
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BibTeX
@misc{raket2026bestiary,
title = {{A Bestiary of Brownian Processes: A Field Guide to Boundary Conditions and Conditioning}},
author = {Lars Lau Raket},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33747},
eprint = {2610.33747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33747},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - A Bestiary of Brownian Processes: A Field Guide to Boundary Conditions and Conditioning
AU - Lars Lau Raket
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33747
UR - https://arxiv.org/abs/2610.33747
AB - The covariance of Brownian motion is the Green kernel of the one-dimensional Laplacian with a pinned left endpoint and a free right endpoint. This correspondence between Green's functions for the Laplacian on an interval and Brownian covariance functions turns out to be a general phenomenon. We use this correspondence to organize Brownian-derived Gaussian processes by three operations: choosing a self-adjoint realization, recentering a path, and conditioning a fixed law. Local Robin and selected nonlocal boundary conditions yield a catalogue of twelve processes with closed-form kernels, Brownian-path constructions, spectra and traces. The periodic and Neumann-Neumann realizations annihilate the constants; projecting the source and imposing zero time average selects the corresponding covariance pseudoinverses. Recentering and conditioning can both give a Brownian path zero time average, but the resulting laws differ: a constraint alone does not specify a process. A rank-one conditioning identity gives a scale-invariant criterion for the covariance effect of a shrinking family of observations. For Brownian finite differences of any fixed positive order, the covariance effect vanishes uniformly as the spacing tends to zero. For m-fold integrated Brownian motion, derivative constraints of orders 1,…,m remain detectable at interior times. For simulation, we identify which grid constructions are exact and which are approximate.
ER -