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In this paper we shall prove the weak convergence of the associated diffusionprocesses of regular subspaces with monotone characteristic sets for a fixedDirichlet form. More precisely, given a fixed 1-dimensional diffusion processand a sequence of its regular subspaces, if the characteristic sets of regularsubspaces are decreasing or increasing, then their associated diffusionprocesses are weakly convergent to another diffusion process. This is anextended result of [13].

Weak convergence of regular Dirichlet subspaces