We construct a Lévy stochastic area for Brownian motion on the Sierpinski gasket. The standard approach via Itô integrals fails because this diffusion has sample paths which are far rougher than those of semimartingales. We thus provide an example demonstrating the restrictions of the semimartingale framework. As a consequence of the existence of the area one has a stochastic calculus and can solve stochastic differential equations driven by Brownian motion on the Sierpinski gasket.
"Stochastic area for Brownian motion on the Sierpinski gasket." Ann. Probab. 26 (1) 132 - 148, January 1998. https://doi.org/10.1214/aop/1022855414