By E. S. Venttsel'

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Of latter relation can be written as ((−L + α − α)Gα u, v) = (u, v) − (αGα u, v), (9) Theory of Dirichlet forms and applications 25 where we used Gα = (α − L)−1 . The relation involving Eα then follows from (7)-(9). For the limit relation we use (7), the relation just shown for Eα to get (u, v) = E(Gα u, v) + α(Gα u, v) hence α(u, v) = E(αGα u, v) + α2 (Gα u, v), and the fact that αGα → 1 as α → +∞. Remark 21. t. 1. ]0, ∞[⊂ ρ(L) 2. α(α − L)−1 ≤ 1 Gα := L := α − G−1 α ✠ (α − L)−1 , α>0 ✒ 26 Sergio Albeverio Table 2 ?

Assuming (24) we have then that ◦ Lµ,k ≡ ∂ ∂2 + βµ,k ∂k 2 ∂k ◦ is a well deﬁned linear operator with D(Lµ,k ) = F Cb∞ , symmetric and such that ◦ ◦ E µ,k (u, v) = (u, (−Lµ,k )v) (with ( , ) the L2 (µ)-scalar product). From Proposition 11 in Chapter 2 we ◦ see that E µ,k is closable in L2 (µ). ◦ ¯◦ The next question is whether the closure Eµ,k = E µ,k of E µ,k in L2 (µ) is a Dirichlet form. This is proven similarly as in the ﬁnite dimensional case. In fact let Φε be an ε-approximation of the unit contraction which is C ∞ on R.

396], [277], [290]). For u ∈ F Cb∞ (E), z ∈ E, k ∈ K ⊂ E (K a linear subspace of E) we deﬁne the (Gˆ ateaux-)derivative in the direction k by: d ∂u (z) ≡ u(z + sk)|s=0 ∂k ds We deﬁne m ∂i f (< z, l1 >, · · · , < z, lm >) < k, li > . = i=1 ◦ E µ,k (u, v) ≡ for u, v ∈ F Cb∞ . ). As before the ﬁrst question to be ◦ answered is: for which µ is E µ,k deﬁned on F Cb∞ closable? t. ∂u vdµ = − ∂k u ∂v dµ − ∂k uvβµ,k dµ, ∀u, v ∈ F Cb∞ . (24) Theory of Dirichlet forms and applications 39 Remark 32. This is an “integration by parts” formula, in the spirit of those in the theory of smooth measures [205], [59], [61], [62], [65], [119] and in “Malliavin calculus” (originally on Wiener space, see [372], [396]).