Winding of planar gaussian processes

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 D = R∞ 0 dsC ′(s)2/(C(0)2 − C(s)2). Correlations of exp(inφt) with integer n, the distribution of the angular velocity φ̇t, 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 1 2 (ln t)2, with proper generalizations to the various classes of fractional Brownian motion. These results are tested numerically. Non integer n is studied numerically. ar X iv :0 90 4. 05 82 v1 [ co nd -m at .s ta tm ec h] 3 A pr 2 00 9 Winding of planar gaussian processes 2