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A polynomial expansion for Brownian motion and the associated fluctuation process

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We start by deriving a polynomial expansion for Brownian motion expressed in terms of shifted Legendre polynomials by considering Brownian motion conditioned to have vanishing iterated time integrals of all orders. We further discuss the fluctuations for this expansion and show that they converge in finite dimensional distributions to a collection of independent zero-mean Gaussian random variables whose variances follow a scaled semicircle. We then link the asymptotic convergence rates of approximations for Brownian L\’evy area which are based on the Fourier series expansion and the polynomial expansion of the Brownian bridge to these limit fluctuations. We close with a general study of the asymptotic error arising when approximating the Green’s function of a Sturm-Liouville problem through a truncation of its eigenfunction expansion, both for the Green’s function of a regular Sturm-Liouville problem and for the Green’s function associated with the classical orthogonal polynomials.

This talk is part of the Probability series.

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