The impact of a random metric upon a diffusing particle

We show that if the singularity of the covariance of the random metric is \(\vert \textbf{x}\vert ^{-2\gamma }\) ${|\mathbf{x}|}^{-2\gamma }$ then the mean value of the fourth power of the distance achieved in time t by a diffusing particle behaves as \(t^{2(1-\gamma )}\) $t^{2(1-\gamma )}$ for a small t.