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SUMMARY:A polynomial expansion for Brownian motion and the associated fluc
 tuation process - Karen Habermann (Warwick)
DTSTART:20220905T140000Z
DTEND:20220905T150000Z
UID:TALK178460@talks.cam.ac.uk
CONTACT:Jason Miller
DESCRIPTION:We start by deriving a polynomial expansion for Brownian motio
 n expressed in terms of shifted Legendre polynomials by considering Browni
 an motion conditioned to have vanishing iterated time integrals of all ord
 ers. We further discuss the fluctuations for this expansion and show that 
 they converge in finite dimensional distributions to a collection of indep
 endent zero-mean Gaussian random variables whose variances follow a scaled
  semicircle. We then link the asymptotic convergence rates of approximatio
 ns for Brownian L\\'evy area which are based on the Fourier series expansi
 on and the polynomial expansion of the Brownian bridge to these limit fluc
 tuations. We close with a general study of the asymptotic error arising wh
 en approximating the Green's function of a Sturm-Liouville problem through
  a truncation of its eigenfunction expansion\, both for the Green's functi
 on of a regular Sturm-Liouville problem and for the Green's function assoc
 iated with the classical orthogonal polynomials.
LOCATION:MR9\, Centre for Mathematical Sciences
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