Let \((B, H, \mu)\) be an abstract Wiener space and \({\mathcal D}^1_2\) denotes the Sobolev space in the sense of S. Watanabe [“Lectures on stochastic differential equations and Malliavin calculus”, Berlin, Heidelberg, New York (1984; Zbl 0546.60054)]. Let \(F\in {\mathcal D}^1_2\) and assume \(e^F\in L^{\infty-} (B, \mu)\), where \(L^{\infty -}(B, \mu)= \bigcap_{p>1} L^p (B, \mu)\). Denote by \[ \begin{aligned} {\mathcal {FC}}_b^\infty (B)= \{&f: \mathbb{B}\to \mathbb{R} \text{ there exist }n\in \mathbb{N} \text{ and }\widetilde {f}\in C_b^\infty (\mathbb{R}) \text{ such}\\ &\text{ that }f(x)= \widetilde {f} (\langle x,e_1 \rangle, \dots, \langle x,e_n \rangle)\} \end{aligned} \] and \(\{e_i\} (\subset B^*)\) stands for an orthonormal basis of \(H\).
Consider the symmetric bilinear form in \(L^2 (\mu_F)\) \[ {\mathcal E}_F (f,g)= \int_B (Df(x), Dg(x) )_{H^*} d\mu_F(x), \] where \(f\), \(g\in{\mathcal {FC}}_b^\infty (B)\) and \(d\mu_F= e^{2F} d\mu/ \int e^{2F} d\mu\). In the above, \(Df\) denotes an \(H\)-derivative in the \(L^2\) sense.
In this paper the authors only consider Markovian extension of \({\mathcal E}_F\), moreover the authors also write the Dirichlet form as \({\mathcal E}_F\) and the generator as \(L_F\). The authors prove the following main theorem:
Assume \(\exp (|DF |^2_H)\in L^{\infty-} (B,\mu)\). Then \(e^F\in L^{\infty-} (B, \mu)\) and
(i) there exists a \(\lambda>0\) such that for \(f\in {\mathcal {FC}}_b^\infty (B)\), \[ \int_B f^2 \log (f^2/|f|^2_{L^2 (\mu_F)} )d\mu_F\leq\lambda{\mathcal E}_F (f,f), \] (ii) \(L_F\) has only discrete spectrum in \([0,1)\).