An example of regular \((r,p)\)-capacity and essential self-adjointness of a diffusion operator in infinite dimensions
On the separable metric space \(X\) let be given the finite Borel measure \(m\) and on the space \(L^p (X, m)\), \(1\leq p< \infty\), the symmetric contraction semigroup \((T_t)\). By \(F_{r, p}\) be denoted the range of \(L^p (X, M)\) with respect to the Gamma transform \(V_r\) of \((T_t)\). By means of \(F_{r, p}\) a so-called \((r, p)\)-capacity \(C_{r, p}\) is defined and asked under which assumption this capacity satisfies the regularity condition \[ F_{r, p}\cap C_b (X) \qquad \text{is dense in } F_{r,p}. \tag{R} \] The purpose of the paper is to give examples satisfying the condition (R). Starting with an abstract Wiener space \((B, H, \mu)\) \(p=2\) is chosen, a function \(\rho\in W^{\infty, -\infty}\) with \(\rho> 0\) \(\mu\)-a.e. is fixed and on \(L^2 (\rho^2 \mu)\) the Dirichlet form \[ E(f, g)= \int_B (Df (x), Dg (x) )_{H^*} \rho (x)^2 \mu (dx) \] (\(Df\) the \(H\) derivative) is introduced. Then it is shown that the associated semigroup \((T_t)\) determines a regular capacity \(C_{r,2}\). In the case \(r= p= 2\) the condition (R) is closely related to essential self-adjointness of the generator \(G\) of \((T_t)\). In this case more general Dirichlet forms are considered, namely \[ E(f, g)= \int_B (\sqrt {A^*} Df(x), \sqrt {A^*} Dg(x))_{H^*} \rho (x)^2 \mu (dx) \] (\(A^*\) a strictly positive definite self-adjoint operator). The standard Ornstein-Uhlenbeck process and the Ornstein-Uhlenbeck process with a general drift play an important role for the derivation of the results.
scientific article; zbMATH DE number 871814