The framework of white noise analysis [\textit{T. Hida}, Brownian motion (1980; Zbl 0432.60002)] is used to construct and investigate Dirichlet forms [\textit{M. Fukushima}, Dirichlet forms and Markov processes. (1980; Zbl 0422.31007)] over \({\mathcal S}^*({\mathbb{R}})\) (the generalization of \({\mathcal S}^*({\mathbb{R}}^ d)\) being obvious). Let (\({\mathcal S}^*({\mathbb{R}}),{\mathcal B},d\mu)\) be the probability space of white noise. With the help of the second quantized Hamiltonian of the harmonic oscillator one constructs a nuclear triple \[ (1)\quad ({\mathcal S}^*\supset L^ 2({\mathcal S}^*({\mathbb{R}}),d\mu)\supset ({\mathcal S}). \] The space (\({\mathcal S})\) of test functionals on \({\mathcal S}^*({\mathbb{R}})\) is an algebra. A gradient \(\nabla:({\mathcal S})\to l^ 2\otimes ({\mathcal S})\) is defined by means of the isomorphy \(L^ 2({\mathcal S}^*({\mathbb{R}}),d\mu)\) and the symmetric Fock space over \(L^ 2({\mathbb{R}})\) and Fréchet differentiation. For \(F\in ({\mathcal S})\) we have \(\| \nabla F\|_{l^ 2}\in ({\mathcal S}).\) By \textit{Y. Yokoi}'s theorem [Positive generalized Brownian functionals. Kumamoto Preprint (1987)] positive elements \(\Phi\in ({\mathcal S})^*\) are represented by a measure \(d\nu\) on (\({\mathcal S}^*({\mathbb{R}}),{\mathcal B}):\) \[ (2)\quad =\int \tilde Fd\nu, \] where \(\tilde F\) is the unique (strong-*) continuous version of \(F\in ({\mathcal S})\). We set for \(F\in ({\mathcal S})\) \[ (3)\quad {\mathcal S}(F)==\int (\| \nabla F\|^ 2_{l^ 2})^{\sim}d\nu. \] The main results of the article are two closability criteria for the form \({\mathcal S}\) on \(L^ 2({\mathcal S}^*({\mathbb{R}}),d\nu)\) and the Markovian contraction property of the closures of the form (3).