Qian Hong, Raymond Gary M, Bassingthwaighte James B
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
J Phys A Math Gen. 1998;31(28):L527. doi: 10.1088/0305-4470/31/28/002.
As a generalization of one-dimensional fractional Brownian motion (1dfBm), we introduce a class of two-dimensional, self-similar, strongly correlated random walks whose variance scales with power law N(2) (H) (0 < H < 1). We report analytical results on the statistical size and shape, and segment distribution of its trajectory in the limit of large N. The relevance of these results to polymer theory is discussed. We also study the basic properties of a second generalization of 1dfBm, the two-dimensional fractional Brownian random field (2dfBrf). It is shown that the product of two 1dfBms is the only 2dfBrf which satisfies the self-similarity defined by Sinai.
作为一维分数布朗运动（1dfBm）的推广，我们引入了一类二维、自相似、强相关的随机游走，其方差按幂律N(2)(H)缩放（0 < H < 1）。我们报告了在大N极限下其轨迹的统计尺寸和形状以及片段分布的分析结果。讨论了这些结果与聚合物理论的相关性。我们还研究了1dfBm的第二种推广形式——二维分数布朗随机场（2dfBrf）的基本性质。结果表明，两个1dfBm的乘积是唯一满足西奈定义的自相似性的2dfBrf。
