一维Brown运动在其正则Dirichlet扩张中的正交补

The orthogonal complements of H^1(R)in its regular Dirichlet extensions

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.