Summary
In the situation of the classical mean motion, we haven planets moving in the plane, planetk+1 being a satellite of planetk. A classcal result then states that planetn has a mean motion,i.e. its mean angular speed between time 0 and timet has a limit whent→∞. We show in this article that any real gaussian dynamical system can be interpreted as the limit of this situation, whenn→∞. From a given nonatomic probability measure σ on [0,π], we construct a transformationT of the complex brownian path (B u)0≤u≤1 which preserves Wiener measure.T is defined as the limit of a sequenceT n, whereT n acts as the motion of 2n planets. In this way we get a real gaussian dynamical system, whose spectral measure is the symetric probability on [-π,π] obtained from σ. The transformationT can be inserted in a flow (T t) t∈ℝ, and the “orbits”t↦Z t=B 1○T t still have almost surely a mean motion, which is the mean of σ.
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de la Rue, T. Mouvement moyen et système dynamique gaussien. Probab. Theory Relat. Fields 102, 45–56 (1995). https://doi.org/10.1007/BF01295220
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DOI: https://doi.org/10.1007/BF01295220
Mathematics Subject Classification
- 28D99
- 60J65