  In complex systems with fractal properties the scale invariance has an
important rule to classify different statistical properties. In two dimensions
the Loewner equation can classify all the fractal curves. Using the
Weierstrass-Mandelbrot (WM) function as the drift of the Loewner equation we
introduce a large class of fractal curves with discrete scale invariance (DSI).
We show that the fractal dimension of the curves can be extracted from the
diffusion coefficient of the trend of the variance of the WM function. We argue
that, up to the fractal dimension calculations, all the WM functions follow the
behavior of the corresponding Brownian motion. Our study opens a way to
classify all the fractal curves with DSI.
  In particular, we investigate the contour lines of 2D WM function as a
physical candidate for our new stochastic curves.
