A Constructive Brownian Limit Theorem
Yuen-Kwok Chan

TL;DR
This paper provides a constructive proof of a boundary limit theorem for Brownian motions in Hardy spaces on the unit ball, extending previous results to the case p=1 and suggesting broader applications.
Contribution
It offers the first constructive proof for the boundary limit theorem in Hardy spaces for p=1, generalizing prior work and enabling potential constructive proofs of nontangential limits.
Findings
Constructive proof for p=1 Hardy space boundary limits.
Extension of constructive proofs to all p ≥ 1.
Potential for constructive proofs of nontangential limit theorems.
Abstract
In this paper, we present and prove a boundary limit theorem for Brownian motions for the Hardy space of harmonic functions on the unit ball in , where and are arbitrary. Our proof is constructive in the sense of [Bishop and Bridges 1985, Chan 2021, Chan 2022]. Roughly speaking, a mathematical proof is constructive if it can be compiled into some computer code with the guarantee of exit in a finite number of steps on execution. A constructive proof of said boundary limit theorem is contained in [Durret 1984] for the case of . In this article, we give a constructive proof for , which then implies, via the Lyapunov's inequality, a constructive proof for the general case . We conjecture that the result can be used to give a constructive proof of the nontangential limit theorem for Hardy spaces with . We…
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Taxonomy
TopicsStochastic processes and financial applications · Advanced Harmonic Analysis Research · advanced mathematical theories
