Exit times for elliptic diffusions and BMO
Exit times for elliptic diffusions and BMO
复制标题
椭圆扩散和 BMO 的退出时间
DOI：
10.1017/s0013091500028339
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发表时间：
1987
影响因子：
0.7
通讯作者：
B. Øksendal
中科院分区：
文献类型：
--
作者：
R. Bañuelos;B. Øksendal
在1948年P.莱维制定了以下定理：如果U是一个开子集的复平面和f：U →是一个非常数的解析函数，那么f映射一个二维布朗运动Bt（直到退出时间从U）到一个时间改变的二维布朗运动。这一结果的严格证明首先出现在McKean [22]。这个定理已被许多作者用来解决问题的解析函数减少他们的问题布朗运动的论点往往是更透明的。调查文件[8]是其中一些应用程序的良好参考。Lévy定理首先由Bernard、坎贝尔和Davie [5]推广，随后由Csink和Schlaksendal [7]推广。在本文的第一节中，我们利用Lévy定理的推广，将单位圆盘上的BMO函数的某些结果推广到Riemann曲面上的调和态射、Riemann曲面上的全纯函数和Riemann曲面上的解析函数。在第二节中，我们刻画了具有椭圆扩散的期望出口时间作为起始点的函数一致有界的性质的区域。这推广了Hayman和Pommerenke [15]以及Stegenga [24]关于复平面上BMO域的一个结果。
In 1948 P. Lévy formulated the following theorem: If U is an open subset of the complex plane and f:U → ℂ is a nonconstant analytic function, then f maps a 2-dimensional Brownian motion Bt (up to the exit time from U) into a time changed 2-dimensional Brownian motion. A rigorous proof of this result first appeared in McKean [22]. This theorem has been used by many authors to solve problems about analytic functions by reducing them to problems about Brownian motion where the arguments are often more transparent. The survey paper [8] is a good reference for some of these applications. Lévy's theorem has been generalized, first by Bernard, Campbell, and Davie [5], and subsequently by Csink and Øksendal [7]. In Section 1 of this note we use these generalizations of Lévy's theorem to extend some results about BMO functions in the unit disc to harmonic morphisms in ℝn to holomorphic functions in ℂn and to analytic functions on Riemann surfaces. In Section 2, we characterize the domains in ℝn which have the property that the expected exit time of elliptic diffusions is uniformly bounded as a function of the starting point. This extends a result of Hayman and Pommerenke [15], and Stegenga [24] about BMO domains in the complex plane.