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Martingale decomposition of Dirichlet processes on the Banach space \(C_ 0[0,1]\). (English) Zbl 0879.60048

Let \(E=C_0[0,1]\) be the Banach space of all continuous functions \(f:[0,1]\to\mathbb{R}\) with \(f(0)=0\) and let \(\mu\) be a Borel probability on \(E\) with \(\operatorname{supp}(\mu)=E\). Denote by \(H\) the classical Cameron-Martin space, \(H=\{f\in W^{1,2}(0,1): f(0)=0\}\). Let \(({\mathcal E},D(\mathcal E))\) be a symmetric Dirichlet form on \(L^2(E,\mu)\) given as a closure of a form defined for smooth cylindrical functions \(u\), \(v\) on \(E\) by \[ {\mathcal E}(u,v)=\frac 12 \int_E \bigl\langle A(z)\nabla u(z),\nabla v(z)\bigr\rangle_{H} d\mu(z), \tag \(*\) \] where \(A:E\to{\mathcal L}(H)\) is a strongly measurable mapping such that \(A(z)\) is self-adjoint and \(\delta I_{H}\leq A(z)\leq\delta^{-1}I_{H}\) in the sense of quadratic forms for a constant \(\delta>0\) and any \(z\in E\). By \(\nabla u(z)\) the unique element in \(H\) representing the linear mapping \(k\to \frac {\partial u}{\partial k}(z)\) is denoted, \(\frac{\partial u}{\partial k}\) standing for the derivative in the direction \(k\). There exists a diffusion process \(((X_{t}),(P_{z})_{z\in E})\) associated with the Dirichlet form \(({\mathcal E},D(\mathcal E)\); set \(P_\mu=\int_{E} P_{z} d\mu(z)\).
The authors prove that the Dirichlet process \(X\) can be decomposed into a forward and a backward martingale under the probability \(P_\mu\). Namely: assume that the dual space \(E^{*}\subseteq L^{2}(E,\mu)\) and set \({\mathcal F}_{t}=\sigma(X_{s}, s\leq t)\), \(\overline{\mathcal F}_{t}=\sigma(X_{1-s}, s\leq t)\). Then there exist an \(E\)-valued continuous \(({\mathcal F}_{t})\)-martingale \(M_{t}\) and an \(E\)-valued continuous \((\overline{\mathcal F}_{t})\)-martingale \(\overline M_{t}\) such that \(X_{t}-X_{0}=\frac 12 M_{t} +\frac 12(\overline M_{1} -\overline M_{1-t})\), \(0\leq t\leq 1\), \(P_\mu\)-almost surely. Moreover, \(M_{t},\overline M_{t}\in L^{p}(P_\mu ;E)\) for every \(p\geq 1\). Previously, such a martingale decomposition had been known to hold only componentwise, for any of the processes \(l(X_{t})\), \(l\in E^{*}\) [see T. J. Lyons and W. Zheng, Astérisque 157-158, 249-271 (1988; Zbl 0654.60059), or T. J. Lyons and T. S. Zhang, Ann. Probab. 22, No. 1, 494-524 (1994; Zbl 0804.60044)]. Finally, a criterion for tightness of a sequence of laws of diffusion processes associated with Dirichlet forms of the type \((*)\) is established.
Reviewer: J.Seidler (Praha)

MSC:

60G44 Martingales with continuous parameter
31C25 Dirichlet forms
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J25 Continuous-time Markov processes on general state spaces
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References:

[1] Albeverio, S.; Kusuoka, S.; Röckner, M., On partial integration in infinite dimensional space and applications to Dirichlet forms, J. London Math. Soc., 42, 122-136 (1990) · Zbl 0667.31010
[2] Albeverio, S.; Röckner, M., Classical Dirichlet forms on topolgical vector spaces - closability and a Cameron-Matin formula, J. Funct. Anal., 88, 395-436 (1990) · Zbl 0737.46036
[3] Albeverio, S.; Röckner, M., Stochastic differential equations in infinite dimensions, solutions via Dirichlet forms, Probab. Theory Related Fields, 89, 347-386 (1991) · Zbl 0725.60055
[4] Ethier, S.; Kurtz, T., Markov Processes (1986), Wiley: Wiley New York
[5] Fukushima, M., Dirichlet forms and Markov Processes (1980), North-Holland: North-Holland Amsterdam, Oxford, New York · Zbl 0422.31007
[6] Lyons, T. J.; Zhang, T. S., Decomposition of Dirichlet processes and its applications, Ann. Probab., 22, 494-524 (1994) · Zbl 0804.60044
[7] Lyons, T. J.; Zheng, W. A., A crossing estimate for the canonical process on a Dirichlet space and a tightness result, Colloque Paul Lévy sur les processes stochastiques, Asterisque, 157-158, 249-272 (1988)
[8] Ma, Z. M.; Röckner, M., An Introduction to the Theory of (Non-symmetric) Dirichlet Forms (1992), Springer: Springer Berlin
[9] McKean, H. P., Stochastic Integrals (1969), Academic Press: Academic Press New York · Zbl 0191.46603
[10] Röckner, M.; Zhang, T. S., Decomposition of Dirichlet processes on Hilbert space, (Barlow, M. T.; Bingham, N. H., Stochastic Analysis (1991), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 321-332 · Zbl 0747.60057
[11] Röckner, M.; Zhang, T. S., Finite dimensional approximation of diffusion processes on infinite dimensional state spaces, (Stochastics and Stochastic Rep. (1995), Fakultät für Mathematik, Universität Bielefeld: Fakultät für Mathematik, Universität Bielefeld Bielefeld), To appear in · Zbl 0885.60066
[12] Schmuland, B., An alternative compacification for classical Dirichlet forms on topological vector spaces, Stochastics, 33, 75-90 (1990) · Zbl 0726.31008
[13] Takeda, M., On a martingale method for symmetric diffusion processes and its applications, Osaka J. Math., 26, 605-623 (1989) · Zbl 0717.60090
[14] Triebel, H., Interpolation theory, function spaces, differential operators (1978), North-Holland: North-Holland Amsterdam · Zbl 0387.46033
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