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On Dirichlet processes associated with second order divergence form operators. (English) Zbl 0974.60064

Let \((X_t,P^x)\) be the diffusion process on \(R^d\) associated with the differential generator \(L^{a,b}={1\over 2} \partial_j(a^{ij}(x)\partial_i) + b^j(x)\partial_j\) for symmetric uniformly elliptic measurable functions \((a^{ij})\) and bounded measurable functions \((b_i)\). Put \(\overline{X}_t=X_{T-t}\), \({\mathcal F}_t=\sigma(X_s;s\leq t)\) and \(\overline{\mathcal F}_t = \sigma(\overline{X}_s;s\leq t)\). In the previous paper [Stochastic Processes Appl. 63, No. 1, 11-33 (1996; Zbl 0870.60073)], the author has given, for a quasi-continuous function \(\varphi\) belonging to the Sobolev space \(W^1_q(R^d)\), a decomposition \(\varphi(X_t)-\varphi(X_0) =M^{x,\varphi}_t+A^{x,\varphi}_t\), into \(({\mathcal F}_t,P^x)\)-martingale \(M^{x,\varphi}_t\) and \({\mathcal F}_t\)-adapted 0-quadratic variation process \(A^{x,\varphi}_t\). Further, \(A^{x,\varphi}_t\) is explicitly given by using a process of integrable variation, \(({\mathcal F}_t,P^x)\)-martingale \(M^{x,\varphi}_t\) and \((\overline{\mathcal F}_t,P^x)\)-martingale \(N^{x,\varphi}_t\). In this paper, for a continuous function \(\varphi \in W^1_q(R^d)\) with \(q>d\vee 2\), the existence of a.e. \(x\)-independent versions \(M^\varphi\), \(A^\varphi\) and \(N^\varphi\) of \(M^{x,\varphi}\), \(A^{x,\varphi}\) and \(N^{x,\varphi}\), respectively, is shown. In the case of \(b=0\) and \(\varphi \in W^1_2(R^d)\), the decomposition is identified with Fukushima’s decomposition. Further, a refinement is given of Lyons-Zheng’s decomposition for q.e. starting points.

MSC:

60J60 Diffusion processes
60H05 Stochastic integrals
31C25 Dirichlet forms

Citations:

Zbl 0870.60073
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