The orthogonal complements of H¹(ℝ) in its regular Dirichlet extensions

2016/11/21 by Yuncong Shen, Liping Li, Shen, Yuncong +3
Economics, Econometrics and Finance · Mathematics · #31C25 #60J55 #60J60 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #advanced mathematical theories #math.PR #msc:31C25 #msc:60J55 #msc:60J60

paper · pdf · doi:10.48550/arxiv.1611.06782

arxiv created 2016/11/21 · openalex publication_date 2016/11/21 · arxiv updated 2016/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider the regular Dirichlet extension (E,F) for one-dimensional Brownian motion, that H¹(ℝ) is a subspace of F and E(f,g)=\frac12D(f,g) for f,g∈ H¹(ℝ). Both H¹(ℝ) and F are Hilbert spaces under E_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 another 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.

Citations

• Regular Dirichlet subspaces and Mosco convergence

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