Journal of the
Korean Mathematical Society JKMS

The fractal space is often associated with natural phenomena with many length scales and the functions defined on this space are usually not differentiable. First we define a $\sigma$-multifractal from $\sigma$-iterated function systems with probability. We introduce the measure derivative through the invariant measure of the $\sigma$-multifractal. We show that the non-differentiable function on the $\sigma$-multifractal can be differentiable with respect to this measure derivative. We apply this result to some examples of ordinary differential equations and diffusion processes on $\sigma$-multifractal spaces.

Keywords: $\sigma$-multifractal, measure derivative, diffusion process, Brownian motion

MSC numbers: 28A80, 58G32, 60J18, 60J60