Dynkin's formula and large coupling convergence
Journal article, 2005
Let H≥1 be a selfadjoint operator in H, let J be a linear and bounded operator from (D(H1/2), ∥ H1/2) . ∥) to Haux) and for β > 0 let HβJ be the nonnegative selfadjoint operator in H satisfying ∥ (HβJ)1/2 f∥2 = ∥ H1/2 f∥2 + β ∥ Jf∥aux)2, f ∈ D((HβJ)1/2) = D (H1/2). Let D∞J be the limit of the operators DβJ = H-1 - (HβJ)-1. For D∞J the generalized Dynkin's formula D∞J = PH-1 is derived. P is the orthogonal projection in (D(H1/2), H1/2·, H1/2·)) onto the orthogonal complement of the kernel of J. Operator and trace class norms of D∞J - DβJ are given explicitly such that their rate of convergence can be studied exactly. Let E be an m-symmetric regular Dirichlet form in L2(E,m), Γ a closed subset of E with finite capacity and the equilibrium measure of Γ. If H - 1 is the selfadjoint operator associated to E and J : D(E) . L2(E, μ1 is defined by Jf = f̃ μ1-a.e., f ∈ D(E), then ∥ D∞J - DβJ ∥ ≤ 1 / 1+β, β > 0. Here f̃ denotes any quasi-continuous representative of f. Estimate (1) is sharp.