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On weak convergence of diffusion processes generated by energy forms. (English) Zbl 0852.60090

We consider a sequence of forms \[ {\mathcal E} (u,v) = \int_{\mathbb{R}^d} A_n (x) \nabla u(x) \cdot \nabla v(x) \phi^2_n (x)dx,\qquad u, v \in C^\infty_0 (\mathbb{R}^d), \] on \(L^2(\mathbb{R}^d; \phi^2_n(x)dx)\), where \(\phi_n\) are locally bounded functions on \(\mathbb{R}^d\), \(A_n\) are \(d \times d\)-symmetric matrix-valued functions on \(\mathbb{R}^d\) and denote by \(C^\infty_0 (\mathbb{R}^d)\) the totality of all infinitely differentiable functions with compact support. Take strictly positive, bounded functions \(f_n\) with \(\int_{\mathbb{R}^d} f_n \phi^2_n dx = 1\) and denote by \(\{X_y, P^n_x, x \in \mathbb{R}^d\}\) the diffusion processes associated with the forms \({\mathcal E}^n\). We show the weak convergence of the probability measures \(\{P^m_{m_n}\), \(n = 1,2\dots\}\) with \(dm_n = f_n \phi^2_ndx\), when the date \(A_n\), \(\phi^2_n\), \(f_n\) converge a.e. on \(\mathbb{R}^d\), as \(n \to \infty\). In order to obtain the result we need to show the convergence of the processes in the sense of finite-dimensional distribution (F.D.D.) and the tightness of the measures. We adopt the Mosco-convergence technique [U. Mosco, J. Funct. Anal. 123, No. 2, 368-421 (1994; Zbl 0808.46042)] of closed forms to see the F.D.D. convergence. The tightness follows from showing the following volume estimates (so-called Takeda’s tightness criteria): For all \(r > 0\), there is \(c > 0\) such that \[ \sup_n \int_{B_0(r)} \phi^2_n (x) dx \leq \text{exp} (cr^2). \] This is an extension of T. J. Lyons and T. S. Zhang [Bull. Lond. Math. Soc. 25, No. 4, 353-356 (1993; Zbl 0793.31006)]. They assumed a uniform boundedness of \(\phi_n\) on the whole space, while a certain locally boundedness is assumed in this paper.
Reviewer: T.Uemura (Kobe)

MSC:

60J60 Diffusion processes
31C25 Dirichlet forms
60F05 Central limit and other weak theorems
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