Fluctuations in the concentration of Brownian particles in one and two dimensions, or any reasonable measurement of the concentration such as in fluorescence correlation spectroscopy, is shown to be a stochastic fractal with a long tail. Being singular at ω=0, the power spectrum of the fluctuation S(ω)~ω - 1 / 2 for diffusion in one dimension, ~log ω in two dimensions, but non-singular in three dimensions. This discovery provides one simple physical mechanism for possible long-memory fractal behavior, and its implications to various biological processes are discussed.