Abstract:  A process with independent and stationary increments is called a Levy process.  A Brownian motion is a continuous Levy process, but a general Levy process has jumps, thus provides a more general model in applications.  Levy processes may be defined in groups because increments may be defined in terms of the group structure.  For Levy processes in compact Lie groups, the Fourier analysis based on Peter-Weyl Theorem provides a convenient tool for study.  When the Levy process has an L^2 distribution density, it may be expanded into a Fourier series, and this allows  us to determine how fast the Levy process converges to the uniform distribution (that is, the normalized Haar measure).  We may also obtain useful conditions under which the Levy process has an L^2, or even smooth, density.