Paths of Stochastic Processes: a Sudden Turnaround

TL;DR

This paper introduces a novel approach to defining stochastic process paths starting from a covariance function, enabling the construction of non-Gaussian random fields while preserving key properties.

Contribution

It reverses the traditional path definition process by beginning with a covariance function, allowing for explicit path representations and analysis of regularity properties.

Findings

Paths are almost surely not in the original RKHS but in larger spaces.

Regularity properties like continuity are preserved across the construction.

Paths for Matern kernels lie in certain Sobolev spaces, revealing a regularity gap.

Abstract

The commonly accepted definition of paths starts from a random field but ignores the problem of setting joint distributions of infinitely many random variables for defining paths properly afterwards. This paper provides a turnaround that starts with a given covariance function, then defines paths and finally a random field. We show how this approach retains essentially the same properties for Gaussian fields while allowing to construct random fields whose finite dimensional distributions are not Gaussian. Specifically, we start with a kernel $C$ and the associated Reproducing Kernel Hilbert Space ${\cal H}(C)$, and then assign standardized random values to a deterministic orthonormal expansion in ${\cal H}(C)$. This yields paths as random functions with an explicit representation formula. Using Lo\'eve isometry, we prove that pointwise regularity notions like continuity or differentiability hold on functions of ${\cal H}(C)$, paths, and the random field $R_C$ in precisely the same way. Yet, norms of paths as functions behave differently, as we prove that paths are a.s. not in ${\cal H}(C)$, but in certain larger spaces that can partially be characterized. In case of Matern kernels generating Sobolev space $W_2^m(R^d)$, paths lie almost surely in all $W_2^{p}(R^d)$ for $p<m-d/2$, but almost surely not in $W_2^{m-d/2}(R^d)$. This regularity gap between function and paths is explained easily by square summability of expansion coefficients of functions, not of paths. The required orthonormal expansions, well-known in the probabilistic and the deterministic literature, are analyzed and compared with respect to convergence rates.