## Regularity of Gaussian Processes on Dirichlet Spaces

**A 2015 Preprint by
G. Kerkyacharian,
S. Ogawa,
P. Petrushev, and
D. Picard
**

- 2015:06
We are interested in the regularity of centered Gaussian processes $(Z _ x( \omega )) _ {x\in M}$ indexed by compact metric spaces $(M, \rho)$. It is shown that the almost everywhere Besov space regularity of such a process is (almost) equivalent to the Besov regularity of the covariance $K(x,y) = \mathbb{E}(Z _ x Z _ y)$ under the assumption that (i) there is an underlying Dirichlet structure on $M$ which determines the Besov space regularity, and (ii) the operator $K$ with kernel $K(x, y)$ and the underlying operator $A$ of the Dirichlet structure commute. As an application of this result we establish the Besov regularity of Gaussian processes indexed by compact homogeneous spaces and, in particular, by the sphere.