I will try to define a non-trivial stochastic process on the Sierpinski gasket by looking atthe limit of stochastic processes on approximating graphs of cells of the Sierpinski gasket.I start by looking more closely on how the fractal is constructed and define randomwalks on the approximating graphs. Then I will define and examine harmonic functionsin the setting of graphs of cells. Next I will consider expected hitting times of the randomwalks. At last I will try to take the limit of the random walks. I will use an analyticalapproach where the methods of Dirichlet forms will be used. My hope is that the techniquesdeveloped in this thesis will apply to more general fractals than the Sierpinskigasket and give new fruitful results to the theory of diffusions on fractals.