Regularity property of Donsker's delta function
type: Item
instance of: scholarly article
title: Regularity property of Donsker's delta function
zbMATH Open document ID: 0552.60035
DOI: 10.1007/BF01449036
author: Gopinath Kallianpur
author: Hui-Hsiung Kuo
published in: Applied Mathematics and Optimization
publication date: 1984-01-01T00:00:00Z
review text: Let \({\mathcal S}\) be the space of rapidly decreasing smooth functions on \({\mathbb{R}}\) and \({\mathcal S}^*\) its dual space. Let \((L^ 2)^+\) and \((L^ 2)^-\) be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise space \({\mathcal S}^*\) with standard Gaussian measure. The Donsker delta function \(\delta\) (B(t)-x) is in \((L^ 2)^-\) and admits the series representation \[ \delta (B(t)-x)=(2\pi t)^{-1/2}\exp (-x^ 2/2t)\sum^{\infty}_{n=0}(n!2^ n)^{-1}H_ n(x/\sqrt{2t})H_ n(B(t)/\sqrt{2t}), \] where \(H_ n\) is the Hermite polynomial of degree n. It is shown that for \(\Phi\) in \((L^ 2)^+\), \(g_{t,\Phi}(x)\equiv <\delta (B(t)-x),\Phi >\) is in \({\mathcal S}\) and the linear map taking \(\Phi\) into \(g_{t,\Phi}\) is continuous from \((L^ 2)^+\) into \({\mathcal S}\). This implies that for f in \({\mathcal S}^*\), \(f(B(t))\equiv <f,\delta (B(t)-(\cdot))>\) is a generalized Brownian functional and admits the series representation \[ f(B(t))=(2\pi t)^{- 1/2}\sum^{\infty}_{n=o}(n!2^ n)^{-1}<f,\xi_{n,t}>H_ n(B(t)/\sqrt{2t}), \] where \(\xi_{n,t}\) is the Hermite function of degree n with parameter t. This series representation is used to prove the Ito lemma for f in \({\mathcal S}^*\), \[ f(B(t))=f(B(u))+\int^{t}_{u}\partial^*_ sf'(B(s))ds+(1/2)\int^{t}_{u}f''(B(s))ds, \] where \(\partial^*_ s\) is the adjoint of \(\dot B(\)s)-differentiation operator \(\partial_ s\).
Mathematics Subject Classification ID: 60G20
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 3879823
zbMATH Keywords: generalized Brownian functionals
zbMATH Keywords: Hermite polynomial
zbMATH Keywords: Ito lemma
MaRDI profile type: Publication
cites work: Generalized multiple Wiener integrals
cites work: Q3857528
cites work: Q3244654
cites work: Calculus on Gaussian white noise. I
cites work: Q5620934
description: scientific article; zbMATH DE number 3879823