Winding of planar gaussian processes

TL;DR

This paper derives formulas for the winding angle variance of planar Gaussian processes, revealing diffusive behavior for stationary cases and logarithmic growth for processes with stationary increments, supported by numerical tests.

Contribution

It provides the first closed-form expression for the winding angle variance of planar Gaussian processes, including stationary and fractional Brownian motion cases.

Findings

Variance of winding angle is derived explicitly.

Winding angle exhibits diffusion with a specific diffusion coefficient.

For processes with stationary increments, variance grows as (1/2)(ln t)^2.

Abstract

We consider a smooth, rotationally invariant, centered gaussian process in the plane, with arbitrary correlation matrix $C_{t t'}$. We study the winding angle $\phi_t$ 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 $D = \int_0^\infty ds C'(s)^2/(C(0)^2-C(s)^2)$. Correlations of $\exp(i n \phi_t)$ with integer $n$, the distribution of the angular velocity $\dot \phi_t$, 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.