New kind of differential equations, called local fractional differential equations, has been proposed for the first time. They involve local fractional derivatives introduced recently. Such equations appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of Fokker-Planck equation has been derived starting from the Chapman-Kolmogorov condition. Such an equation is solved, with a specific choice of the transition probability, and shown to give rise to subdiffusive behavior.