Winding of planar Gaussian processes

We consider a smooth, rotationally invariant, centered Gaussian process in the plane, with arbitrary correlation matrix C_tt′. We study the winding angle _t, around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrix C _tt′. For most stationary processes C_tt′ = C(t-t′) the winding angle exhibits diffusion at large time with diffusion coefficient . Correlations of exp(in_t) with integer n, the distribution of the angular velocity φ_t , and the variance of the algebraic area are also obtained. For smooth processes with stationary increments (random walks) the variance of the inding angle grows as , with proper generalizations to the various classes of fractional Brownian motion. These results are tested numerically. Non-integer n is studied numerically.