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Brownian motion conditioned to have trivial signature

Banff International Research Station for Mathematical Innovation and Discovery

To motivate of why it could be interesting to study multidimensional Brownian motion conditioned to have trivial signature, we discuss results on one-dimensional Brownian motion on the time interval $ [0, 1] $ conditioned to have vanishing iterated time integrals up to order $N$. We show that, in the large $N$ limit, these processes converge weakly to the zero process, which gives rise to a polynomial decomposition for Brownian motion, and we show that the associated fluctuation processes converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle.

Mathematics; Probability Theory And Stochastic Processes; Global Analysis; Analysis On Manifolds

video/mp4

Author affiliation: University of Warwick

2021-09-08

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/79662

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/