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Extra resources for Lectures on Probability Theory and Statistics. Ecole d'Ete de Probabilites de Saint-Flour XXX, 2000
Sample text
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]). 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).
168). Set A0 = J ∗ . Then we have ∀u ∈ D(J ∗ ), v ∈ H1 : (A0 u, v) = (J ∗ u, v) = (u, Jv)1 = (u, v)1 . Set A = A0 , looked upon as an operator from H into H. It is then clear that D(A) ⊂ H1 ⊂ H, Theory of Dirichlet forms and applications (Au, v) = (A0 u, v) = (u, v)1 ∀u ∈ D(A), v ∈ H1 . 23 (∗) That A ≥ κ follows from the fact that (Au, u) = (u, u)1 ≥ κ(u, u), ∀u ∈ D(A), by the definition of (·, ·)1 . That A is symmetric in H follows from (Au, v) = (u, v)1 , ∀u ∈ D(A), v ∈ H1 and, for v ∈ D(A): (u, Av) = (u, A0 v) = (u, J ∗ v) = (Ju, v)1 = (u, v)1 .
Lµ . Even though ˜◦ E and A in L2 (Rd ) have a more direct physical interpretation, rather than their corresponding objects Eµ resp. g. , [278], [15], [41] (in fact in the latter case there are interesting probability measures µ on E, whereas no good analogue of Lebesgue measure on E exists). In this sense operating with Eµ , Lµ is more natural and general than operating with Schr¨ odinger operators. Remark 26. In the above discussion of closability and Doob’s transformation we have assumed ϕ, V , V˜ to be smooth and ϕ > 0.
