Abstract
We consider a smooth, rotationally invariant, centered Gaussian
process in the plane, with arbitrary correlation matrix
Ctt'. We study the
winding angle ϕt, around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrix
Ctt'. For most
stationary processes Ctt' = C(t−t')
the winding angle exhibits diffusion at large time with diffusion coefficient 
. Correlations of exp(inϕt)
with integer n, the distribution of the angular velocity 
, and the variance of the algebraic area are also obtained. For smooth processes with
stationary increments (random walks) the variance of the winding 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.