When does a Gaussian process have its paths in a reproducing kernel Hilbert space?
Ingo Steinwart
TL;DR
This work characterizes when a Gaussian process $X$ admits a version $Y$ that lies a.s. in some RKHS $H$ on $T$, extending beyond the covariance RKHS $H_X$. A central, general impossibility/possibility dichotomy is established via nuclear embeddings: if $H_X\ll H$ for some $H$ and a version $Y$ exists in $H$, then the embedding $I_{X,\nu}:H_X\to L_2(\nu)$ is nuclear for all finite measures $\nu$; conversely, non-nuclearity for some $\nu$ rules out any such $H$. The paper then delivers complete positive results for Gaussian processes whose $H_X$ is essentially equivalent to fractional Sobolev spaces, showing containment iff the Sobolev order exceeds a dimension-dependent threshold, and providing explicit $H$ choices (e.g., $H^r(T)$ with $d/2<r< s-d/2$). It extends these insights to Sobolev spaces with mixed smoothness and to concrete examples, including Wiener, fractional Brownian motion, Riemann–Liouville processes, Matérn kernels, and translation-invariant fields on groups, across a spectrum of dimensions. Overall, the results sharply delineate when Gaussian-path containment in RKHSs is possible, guiding both theory and kernel-based modeling of Gaussian processes.
Abstract
We investigate for which Gaussian processes there do or do not exist reproducing kernel Hilbert spaces (RKHSs) that contain almost all of their paths. In particular, we establish a new result that makes it possible to exclude the existence of such RKHSs in many cases. Moreover, we combine this negative result with some known techniques to establish positive results. Here it turns out that for many classical families of Gaussian processes we can fully characterize for which members of these families there exist RKHSs containing the paths. Similar characterizations are obtained for Gaussian processes, for which the RKHSs of their covariance functions are Sobolev spaces or Sobolev spaces of mixed smoothness.