论文标题
建设性的布朗极限定理
A Constructive Brownian Limit Theorem
论文作者
论文摘要
在本文中，我们介绍并证明了Brownian动作的边界限制定理，用于Hardy Space $ \ Mathbf {H}^{p} $谐波功能在$ r^m $中的谐波函数的$，其中$ p \ geq1 $和$ m \ geq2 $是任意的。从[Bishop and Bridges 1985，Chan 2021，Chan 2022]的意义上讲，我们的证明具有建设性。粗略地说，如果可以将其编译到某些计算机代码中并保证以有限的执行步骤退出，则数学证明是建设性的。对于$ p> 1 $的情况，[Durret 1984]中包含上述边界限制定理的建设性证明。在本文中，我们给出了$ P = 1 $的建设性证明，然后通过Lyapunov的不平等，这意味着一般情况$ P \ geq1 $的建设性证明。我们猜想结果可以用来为Hardy Spaces的非义务限制定理提供建设性证明$ \ Mathbf {H}^{p} $，并用$ P \ geq1 $。 我们注意到，加利福尼亚州1970年，R。Getoor就华盛顿大学的布朗限制定理发表了演讲。我们认为，他提出的证据仅对案件$ p> 1 $而不是为$ p = 1 $的案例建设性。但是，我们无法找到他的证据参考。
In this paper, we present and prove a boundary limit theorem for Brownian motions for the Hardy space $\mathbf{h}^{p}$ of harmonic functions on the unit ball in $R^m$, where $p\geq1$ and $m\geq2$ 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 $p>1$. In this article, we give a constructive proof for $p=1$, which then implies, via the Lyapunov's inequality, a constructive proof for the general case $p\geq1$. We conjecture that the result can be used to give a constructive proof of the nontangential limit theorem for Hardy spaces $\mathbf{h}^{p}$ with $p\geq1$. We note that, ca 1970, R. Getoor gave a talk on the Brownian limit theorem at the University of Washington. We believe that the proof he presented is constructive only for the case $p>1$ and not for the case $p=1$. We are however unable to find a reference for his proof.
