Fractal diffusion from a geometric Ricci flow

We study a geometric Ricci flow on a fixed manifold where space and time scales followed the two-scale transform (xβ,tα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x^{\beta } ,t^{\alpha } )$$\end{document}, α,β>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ,\beta > 0$$\end{document}. The fractal evolution equations are obtained and the resultant Ricci flow is derived accordingly. For slow velocity of the flow, it was observed that Ricci flow is characterized by an anomalous Gaussian measure similar to what is obtained in anomalous diffusion with variable diffusion coefficient. This result indirectly shows that the fractal Ricci flow may be used to study nonlinear reaction–diffusion processes arising in various fields of applied mathematics.