Some functionals generated by distributions for Brownian motion (Q2777830)

Let \((\Omega,\mathcal F, P)\) be a complete probability space and let \(w(t),t\in [0;1],\) be a Brownian motion on \((\Omega,\mathcal F,P)\). For any \(\alpha\in L^2(\Omega,\mathcal F,P)\) there exists a unique sequence \(\{\varphi_n\in\widehat L^2([0;1]^n),n\geq 0\}\) such that \(\alpha=\sum_{n=0}^{\infty}I_n(\varphi_n)\), where \(I_n(\varphi_n)\) is the multiple Wiener integral with respect to \(w(t)\) and \(\widehat L^2([0;1]^n)\) is the symmetric part of \(L^2([0;1]^n)\). This decomposition is denoted as \(\alpha\sim(\varphi_n)\). For a positive self-adjoint operator \(A\) on \(L^2(R)\) denote by \(\|\alpha\|_{A^p}\) the norm \(\|\alpha\|_{A^p}= :\sum_{n=0}^{\infty}n!\|((A^p)^{\otimes n}\varphi_n)\|_n^2\) generated by \(A\). \(\alpha\) is called a generalized Wiener functional if \(\|\alpha\|_{A^p}<\infty, p>0\). The corresponding space of distributions is denoted by \(S_{-p}^{A}= :\{\alpha\sim(\varphi_n):\|\alpha\|_{A^p}<\infty\}\). The author gives estimates for the degree of smoothness of the functionals NEWLINE\[NEWLINEL(T,\kappa,n)= \int_0^T\left[\kappa(w(t))-\sum_{k=0}^{n-1}I_n(\varphi-k(t))\right]dt, \qquad n\in N,T\in[0;1],NEWLINE\]NEWLINE where \(\kappa(w(t))\) is a functional determined by a distribution \(\kappa\) on the Schwartz space of rapidly decreasing functions on \(R\). If the distribution \(\kappa\) is a Dirac function, then the functional \(L(T,\kappa,n)\) is a version of the local time for the Brownian motion.