Regularity property of Donsker's delta function
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 \(\partial_ s\).
scientific article; zbMATH DE number 3879823