一维Brown运动在其正则Dirichlet扩张中的正交补
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摘要
考虑一维Brown运动的正则Dirichlet扩张(ε,F),即H~1(R)是F的子空间,并且任意的f,g∈H~1(R)满足ε(f,g)=1/2D(f,g).由于H~1(R)和F在ε_α下都是Hilbert空间,因此存在α-正交补g_α.本文给出g_α中函数的具体表达式,它们可以被另两个函数空间刻画.这两个空间上存在自然的广义Dirichlet型,通过补丁变换可以给出它们的正则表示.
Consider the regular Dirichlet extension(ε,F) for one-dimensional Brownian motion, that H~1(R) is a subspace of F and ε(f,g)=1/2 D(f,g) for f,g∈H~1(R). Both H~1(R) and F are Hilbert spaces under ε_α and hence there is α-orthogonal compliment g_α. We give the explicit expression for functions in g_α which then can be described by other two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.
Consider the regular Dirichlet extension(ε,F) for one-dimensional Brownian motion, that H~1(R) is a subspace of F and ε(f,g)=1/2 D(f,g) for f,g∈H~1(R). Both H~1(R) and F are Hilbert spaces under ε_α and hence there is α-orthogonal compliment g_α. We give the explicit expression for functions in g_α which then can be described by other two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the darning method, their regular representations are given.
引文
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8 Li L P,Song X C.The a-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion.Sci China Math,2016,59:2019-2026
9 Fukushima M.Regular representations of Dirichlet spaces.Trans Amer Math Soc,1971,155:455-455
2 Li L,Ying J.Regular subspaces of Dirichlet forms.In:Festschrift Masatoshi Fukushima.Hackensack:World Scientific Publishing,2015,397-420
3 Li L,Ying J.On structure of regular Dirichlet subspaces for one-dimensional Brownian motion.ArXiv:1412.1896,2016
4 Chen Z Q,Fukushima M.Symmetric Markov Processes,Time Change,and Boundary Theory.Princeton:Princeton University Press,2012
5 Fukushima M,Oshima Y,Takeda M.Dirichlet Forms and Symmetric Markov Processes.Berlin:Walter de Gruyter,2011
6 Song X,Li L.Regular Dirichlet subspaces and Mosco convergence.Chin Ann Math Ser A,2016,37:1-14
7 Li L,Ying J.Regular Dirichlet extensions of one-dimensional Brownian motion.ArXiv:1606.00630,2016
8 Li L P,Song X C.The a-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion.Sci China Math,2016,59:2019-2026
9 Fukushima M.Regular representations of Dirichlet spaces.Trans Amer Math Soc,1971,155:455-455