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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.

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