Abstract
We consider a smooth, rotationally invariant, centered Gaussianprocess in the plane, with arbitrary correlation matrixCtt′. We study thewinding angle ϕt, around its center. We obtain a closed formula for the variance of the winding angle as a function of the matrixCtt′. For moststationary 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 withstationary increments (random walks) the variance of the winding angle grows as , with proper generalizations to the various classes of fractionalBrownian motion. These results are tested numerically. Non-integern is studied numerically.
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