论文标题
缩放自我相互作用的随机步行到Brownian运动的融合和不连贯性
Convergence and non-convergence of scaled self-interacting random walks to Brownian motion perturbed at extrema
论文作者
论文摘要
我们使用BálintTóth在1996年引入的广义射线骑士定理，以及开发用于激发随机步行的技术，作为建立正面和负面结果的主要工具，这些工具涉及某些扩散缩放的自我相互作用随机步行（SIRW）与Brownian Motions conterctions cormanta（BMPE）的收敛。 Tóth的作品研究了两类SIRW：渐近自由和多项式自我重复步行。对于这两个类别，尤其表明，在独立的几何时间上观察到的缩放SIRW的分布函数会收敛到该siRW的广义射线骑士定理所指示的BMPE。缩放SIRW的一维分布的弱收敛性问题保持开放。在本文中，一方面，我们证明了一大批渐近免费SIRW的功能性限制定理，其中包括Tóth的论文中考虑的渐近自由步行。另一方面，我们表明，重新恢复的多项式自我重复的SIRW不会融合到相应的广义射线骑士定理所预测的BMPE，因此不会收敛到任何BMPE。
We use generalized Ray-Knight theorems introduced by Bálint Tóth in 1996 together with techniques developed for excited random walks as main tools for establishing positive and negative results concerning convergence of some classes of diffusively scaled self-interacting random walks (SIRWs) to Brownian motions perturbed at extrema (BMPE). Tóth's work studied two classes of SIRWs: asymptotically free and polynomially self-repelling walks. For both classes Toth has shown, in particular, that the distribution function of a scaled SIRW observed at independent geometric times converges to that of a BMPE indicated by the generalized Ray-Knight theorem for this SIRW. The question of weak convergence of one-dimensional distributions of scaled SIRW remained open. In this paper, on the one hand, we prove a full functional limit theorem for a large class of asymptotically free SIRWs which includes asymptotically free walks considered in Tóth's paper. On the other hand, we show that rescaled polynomially self-repelling SIRWs do not converge to the BMPE predicted by the corresponding generalized Ray-Knight theorems and, hence, do not converge to any BMPE.
