Martingale decomposition of Dirichlet processes on the Banach space \(C_ 0[0,1]\)
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 \textit{T. J. Lyons} and \textit{W. Zheng}, Astérisque 157-158, 249-271 (1988; Zbl 0654.60059), or \textit{T. J. Lyons} and \textit{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.
scientific article; zbMATH DE number 1095917
scientific article