Probabilities Estimates For Brownian Process With Polynomial Drift And Time Estimates For Lévy Process On P-Adics

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.