A Bestiary of Brownian Processes: A Field Guide to Boundary Conditions and Conditioning

Source 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 $1,\ldots,m$ remain detectable at interior times. For simulation, we identify which grid constructions are exact and which are approximate.