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Chen, Z. -Q.; Fitzsimmons, P. J.; Kuwae, K.; Zhang, T. -S. (2008)
Publisher: The Institute of Mathematical Statistics
Languages: English
Types: Article
Subjects: 60J55, 60J57, 60H05, additive functional, 31C25, generalized Itô formula, Symmetric Markov process, dual additive functional, Mathematics - Probability, 31C25 (Primary) 60J57, 60J55, 60H05 (Secondary), stochastic integral, Revuz measure, time reversal, martingale additive functional, dual predictable projection
Identifiers:doi:10.1214/07-AOP347
Using time-reversal, we introduce a stochastic integral for zero-energy additive functionals of symmetric Markov processes, extending earlier work of S. Nakao. Various properties of such stochastic integrals are discussed and an It\^{o} formula for Dirichlet processes is obtained.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • Lemma 2.17. Let A be a finite rcll PrAF with defining sets {Ξ, Ξt, t ≥ 0}. There then exists a real-valued 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 m-null 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), Pm-a.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-
    • tion of symmetric Markov processes. Probab. Theory Related Fields 140 239-275. [3] Chen, Z.-Q., Fitzsimmons, P. J., Takeda, M., Ying, J. and Zhang, T.-S. (2004).
    • Absolute continuity of symmetric Markov processes. Ann. Probab. 32 2067-2098.
    • MR2073186 [4] Fitzsimmons, P. J. (1995). Even and odd continuous additive functionals. In Dirich-
    • let Forms and Stochastic Processes (Beijing, 1993 ) (Z.-M. Ma, M. R¨ockner and
    • J.-A. Yan, eds.) 139-154. de Gruyter, Berlin. MR1366430 [5] Fitzsimmons, P. J. and Kuwae, K. (2004). Non-symmetric perturbations of sym-
    • metric Dirichlet forms. J. Funct. Anal. 208 140-162. MR2034295 [6] Fukushima, M. (1980). Dirichlet Forms and Markov Processes. North-Holland, Am-
    • sterdam. MR0569058 [7] Fukushima, M., O¯ shima, Y. and Takeda, M. (1994). Dirichlet Forms and Sym-
    • metric Markov Processes. de Gruyter, Berlin. MR1303354 [8] Getoor, R. K. (1986). Some remarks on measures associated with homogeneous ran-
    • dom measures. In Seminar on Stochastic Processes 1985 (E. C¸ inlar, K. L. Chung,
    • R. K. Getoor and J. Glover, eds.) 94-107. Birkh¨auser, Boston. MR0896738 [9] He, S. W., Wang, J. G. and Yan, J. A. (1992). Semimartingale Theory and Stochas-
    • tic Calculus. Science Press, Beijing. MR1219534 [10] Kuwae, K. (1998). Functional calculus for Dirichlet forms. Osaka J. Math. 35 683-
    • 715. MR1648400 [11] Kuwae, K. (2007). Maximum principles for subharmonic functions via local semi-
    • Dirichlet forms. Canad. J. Math. To appear. [12] Lunt, J., Lyons, T. J. and Zhang, T.-S. (1998). Integrability of functionals of
    • of heat kernels. J. Funct. Anal. 153 320-342. MR1614598 [13] Ma, Z.-M. and Ro¨ckner, M. (1992). Introduction to the Theory of (Non-Symmetric)
    • Dirichlet Forms. Springer, Berlin. [14] Nakao, S. (1985). Stochastic calculus for continuous additive functionals of zero
    • energy. Z. Wahrsch. Verw. Gebiete 68 557-578. MR0772199 [15] Sharpe, M. (1988). General Theory of Markov Processes. Academic Press, Boston,
    • MA. MR0958914 [16] Walsh, J. B. (1972). Markov processes and their functionals in duality. Z. Wahrsch.
    • Verw. Gebiete 24 229-246. MR0329056
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