  In this work we introduce and study fractional measure theoretic elliptic
operators on the torus and a new stochastic process named W-Brownian motion. We
establish some regularity and spectral results related to the operators cited
above, more precisely, we were able to provide sharp bounds for the growth rate
of eigenvalues to an associated eigenvalue problem. Moreover, we show how the
Cameron-Martin space associated to the W-Brownian motion relates to Sobolev
spaces connected with the elliptic operators mentioned above. Finally
applications of the theory developed on stochastic partial differential
equations are given.
