On weak convergence of diffusion processes generated by energy forms
Toshihiro Uemura · Osaka City University (Osaka City University) · 1995
Convergences of closed forms, energy forms or energy functions have been studied by many authors (see e.g. [l]-[3], [5]-[8]). It is important to know, given an energy form, if it can be approximated by nice ones, or, given a sequence of energy forms, what their limit is. In this paper we consider a sequence of forms $\u,v) = \R*(An(x)Vu(x Vv(x))dφ (x)dx with certain domains on L(R\φldx where φn are locally bounded functions on R, An are (dxd) symmetric matrix valued functions on R , ( , )d means the inner product on R and Vu = lu9V2u9 9Vju) is the distributional (weak) derivative of u. Take strictly positive, bounded functions fn with \Rdfnφldx= and denote by {Xt,P n x,x e R ] the diffusion processes associated with the forms S. We study the weak convergence of the probability measures {JPJJtn,« = l,2, } with dmn=fnφ 2 ldx, when the date An, φn9 and fn converge a.e. on R, as n -> oo. Although our main result (see section 1) is similar to that of T.J. Lyons and T.S. Zhang [5], we assume only a certain local boundedness of φn, while a uniform boundedness on the whole space is assumed in [5]. In order to obtain the result in [5], they generalized the theorem of Kato and Simon on monotone sequence of closed forms (see M. Reed and B. Simon [7]) used by S. Albeverio, R. H0egh-Krohn and L. Streit [1]. We will instead adopt the Mosco-convergence of closed forms(see U. Mosco [6]) to prove our theorem. S. Albeverio, S. Kusuoka and L. Streit [2] obtained a semigroup convergence by imposing the regularity conditions that there exist R>0 and C>0 such that, the restrictions of φn to R d — BR is of class C 2 and the growth order of χ-Vφn/φn is not greater than C\x 2 on R — BR. No smoothness on An9 φn is required in the present approach.