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Lemma 2.17. Let A be a finite rcll PrAF with defining sets {Ξ, Ξt, t ≥ 0}. There then exists a realvalued Borel function ϕ on EΔ × EΔ with ϕ(x, x) = 0 for x ∈ EΔ such that A with defining sets
Proof. Let Ξ ∈ F∞, Ξt ∈ Ftm, t > 0 be the defining sets of A admitting mnull set. We easily see rt−1(Ξt) ∩ {t < ζ} ⊂ rs−1(Ξs) ∩ {s < ζ} for s ∈ ]0, t[ by use of Lemma 2.15(i) and θt−sΞt ⊂ Ξs.
Set Ξbt := rt−1(Ξt) for t > 0 and Ξb := Tt>0 Ξbt. We then see that Ξb = Tt>0,t∈Q Ξbt by use of rt−1(Ξt) ∩ {t ≥ ζ} = {t ≥ ζ} and the monotonicity of Lemma 3.1.
Theorem 3.6. For an MAF M of finite energy, Λ(M ) defined above coincides on [[0, ζ[[ with Γ(M ) defined in (1.5), Pma.e. for all t > 0 by way of its subadditivity. Hence, we obtain (3.12).
(iii) Suppose that f ∈ F . Let Kt be a purely discontinuous local MAF on [[0, ζ[[ with Kt − Kt− = −ϕ(Xt−, Xt) − ϕ(Xt, Xt−) on ]0, ζ[. Then, 2 i,j=1 s [2] Chen, Z.Q., Fitzsimmons, P. J., Kuwae, K. and Zhang, T.S. (2008). Perturba
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Absolute continuity of symmetric Markov processes. Ann. Probab. 32 20672098.
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let Forms and Stochastic Processes (Beijing, 1993 ) (Z.M. Ma, M. R¨ockner and
J.A. Yan, eds.) 139154. de Gruyter, Berlin. MR1366430 [5] Fitzsimmons, P. J. and Kuwae, K. (2004). Nonsymmetric perturbations of sym