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Unbiased Shifts for Brownian Motion

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Unbiased shifts of Brownian motion Based on joint work with Günter Last and Peter Mörters

Let B = (B(t) : t in R) be a two-sided standard Brownian motion. Let T be a real-valued measurable function of B. If T is a nonnegative stopping time then the shifted process (B(T + t) – B(T) : t nonnegative) is a one-sided Brownian motion independent of B(T). However, the two-sided process (B(T + t) – B(T) : t in R) need not be a Brownian motion. Moreover, the example of a fixed time T = s shows that even if it is, it need not be independent of B(T).

Call T an unbiased shift of B if (B(T + t) – B(T) : t in R) is a Brownian motion independent of B(T). Unbiased shifts can be characterized in terms of allocation rules balancing additive functionals of B. For any probability distribution Q on R we construct a stopping time T with the above properties such that B(T) has distribution Q. Also moment and minimality properties of unbiased shifts are given.

The case when Q is concentrated at zero is of special interest. We obtain a rigorous formulation of the intuitive idea that B looks globally the same from all its zeros, thus resolving an issue raised by Mandelbrot in The Fractal Geometry of Nature. The result can be stated as follows: if we travel in time according to the clock of local time we always see a two-sided Brownian motion.

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