【作者】 焦丽；

【导师】 应坚刚；

【作者基本信息】 复旦大学 ， 概率论与数理统计， 2007， 博士

【摘要】 本文前大半部分主要研究了期望不为零的Gaussian过程,在时间区间变化时的水平通过概率以及大小偏差概率的估计问题.因为相应于期望为零的某些特殊的Gaussian过程(例如过程本身平稳的Gaussian过程,或者过程增量平稳的Gaussian过程)的水平通过概率以及大小偏差概率的估计问题已经被很多研究者研究过,并有很多的显示结果给出了.但是对于期望不为零的Gaussian过程,却很少有这类的显示结果给出过.因而本文就Levy Gaussian过程,i.e.带定向漂移的Brownian运动以及带“多项式漂移项”———“f(t)=ακtκ+ακ-1tκ-1+…+α1t”的Brownian运动,这两类特殊的过程,给出了相应于它们的水平通过概率以及大小偏差概率的一些显示估计结果.而在本文的最后一部分作者基于常返性及该等价定理,对支撑有界的Levy测度μ的Levy过程Xμ跑遍支撑球k(0,d)中所有pn-balls所需时间进行了研究,并在一些特殊情况下,给出了该时间的理论刻画,对一般情况给出了相应的估计.本文的具体安排如下：首先在第一章的第一节,作者主要介绍了本文的主要研究模型以及本文所要涉及的一些基本概念.在本章第二节作者简单介绍了Gaussian过程概率估计问题的研究历史与现状,并且概括列述出了本文的主要研究结果.作为本文研究的重点,在第二章作者主要给出了一个重要的引理(它在下面几章的的证明中起到了重要作用),期望为零的Levy Gaussian过程在时间区间变化时([ε,T],ε→0,和[ε,T],T→∞)时的概率估计以及相应的极限结果,以及当期望不等于零时的LevyGaussian过程在时间区间变换时([ε,T],ε→0,和[ε,T],T→∞以及[0,T],T→∞)时的几种水平通过概率估计结果,并给出了一些相应的推广.而在第三章主要给出了期望不为零的Levy Gaussian过程的大小偏差概率的估计问题以及相关的推广.在本文第四章作者主要给出了带“多项式漂移项”———“f(t)=ακtκ+ακ-1tκ-1+…+α1t”的Brownian运动大小偏差概率以及水平通过概率估计的一些极限结果,并给出了它们相关的推论.最后在第五章,在第二节主要介绍了P-adics数域,其中的(5.2.3)式的超度量不等式是非常重要的,并对P-adics上的线性不定方程和多维Poisson过程作了简单介绍作了简单介绍,其在本文的论述中起到了很大的作用.第三节首先介绍了P-adics上的Levy过程与一般实直线上的Levy过程性质的几种主要不同之处,进而介绍了一个对本文相当重要的等价定理.基于以上三节的内容,第四节中对P-adics上的Levy过程,设μ是一Levy测度,且supp(μ)是一有界集,d=sup(‖x‖p：x∈supp(μ))=pM.令X=(Xtμ)t≥0是相应于测度μ的Levy过程,当supp(μ)是有界集时,X是常返的,在supp(μ)中的每一小球能被无穷多次撞击,i.e.对任意球B∈K(0,d),α∈K(0,d),有本章主要要考虑的问题是：(?)n≤M,需要多长时间T,Xtμ才能首次跑遍支撑球K(0,PM)中的所有Pn-小球?在第五章作者主要给出了所需时间T的分布密度以及所需要的平均时间(T的期望),得到了较好的结果.更多还原

【Abstract】 The present Ph.D. thesis deals with the Level crossing probabilities estimates in the variational time intervals for Levy Gaussian process and Brownian motion with polyno-mial drift, and Lower tail and Large deviation Probabilities estimates for Levy Gaussian process and Brownian motion with polynomial drift in the foregoing half part. The Level crossing probabilities estimates in the variational time intervals, and Lower tail and Large deviation Probabilities estimates for some special Gaussian process with mean zero(e.g stationary Gaussian process and stationary increment Gaussian process) have been ex-plored by many researchers. And many obvious results have been given out.But there are few similar results for Levy Gaussian process with mean non-zero, which becomes the explored emphases in this thesis. In the latter half part, on the base of recurrence and equivalence theorem, the author studies estimates of times needed by Levy process Xμon P-adics running all pn-balls in the supporting ball k(0, d)for the first time.The arrangement of the thesis is as follows:First of all in Chapter 1, the author gives researched history and actuality of probabil-ity estimates for Gaussian process, the definition and properties of Levy Gaussian process and Brownian motion with polynomial drift.In Chapter 2, the author gives an important lemma firstly, which is very important in the proofs of the following several Chapters. Then, the author obtains some results of probability estimates in the variational time intervals([ε,T],asε→0; [ε,T],as T→∞) for Levy Gaussian process with mean zero, and several Level crossing probabili-ties estimates results in the variational time intervals([ε, T], asε→0;[ε,T],as T→∞; and [0, T], as T→∞) for Levy Gaussian process with mean non-zero.Chapter 3 is devoted to lower tail and large deviation Probabilities estimates for Levy Gaussian process with mean non-zero and related extension.In Chapter 4, the author mainly establishes several limit results for small and large deviation probabilities and level crossing probabilities estimates for Brownian motion with polynomial drifts. Lastly, the author gives out some related corollaries.At last, in Chapter 5, the author mainly introduces P-adics number fields, the linear indefinite equations and multidimensional Poisson processes, which play an important role in the proofs of theorems. In section 3, firstly the author gives several different properties between Levy process on P-adics and Levy process on general real number field, moreover, gives a greatly important theorem to this chapter. On the base of the two sections above, the author gives out several good results about time estimates for Levy process on P-adics space.更多还原