两类随机过程以及相关问题的研究
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摘要
本文主要用对偶、鞅、矩母函数等方法对排它过程和随机树进行研究.首先,我们用对偶的方法,利用随机游动和Fourier变换等方法研究了排它过程中的占位时函数的方差的上界问题；其次,我们研究了排它过程在随机图上的存在性,利用鞅方法,我们将所研究的问题转化为证明对应于生成元鞅解的存在唯一性问题,从而解决了排它过程在随机图上的存在性问题；再次,我们利用矩母函数研究了去点的树过程的对应的子树的顶点数问题；最后,我们利用矩函数和鞅方法研究了推广的树的度分布,分枝结构,给定顶点的度和最大度问题.
本文主要分成以下七个章节.在第一章,我们主要介绍了所研究问题的历史,现状和本文所做的工作；在第二章中,我们研究了一个变分公式和两类由生成元和准生成元所诱导出的范数的等价关系；在第三章中,利用第二章中的变分公式,我们给出了排它过程的占位时函数的方差的一个上界；第四章,我们主要讨论了图上排它过程的生成元对应的鞅问题的解的存在唯一性；在第五章,我们主要讨论了对于均匀递归树,随机删除一个顶点后,剩余子树的顶点数问题；在第六章中,我们对于均匀递归树和有偏好加点树进行了推广,并研究了在推广的随机树中的度分布,分枝结构和最大度问题.在第七章,我们继续讨论随机递归树,我们考虑一类树,其生成是在偶数时间,有偏好加点,在奇数时间均匀加点或以m为时间间隔,在m的倍数时间我们有偏好加点,在其它时间均匀加点,对于这样的树过程,我们考虑它们的度分布问题.
In this paper, we study the exclusion process and random tree by the methods of dualty, martingale, moment generating function, etc.
Firstly, we study the upper bound of the occupation time variance by some knowledge of random walk and Fourier transform; Secondly, we prove the existence of exclusion process on random graph, we transform the problem into the solution of the martingale problem corresponding to the generator of the exclusion, so the problem of existenc is settled; Thirdly, we study the problem of number of vertices in nodes deletion tree process by the means of moment generating function; Finally, by the means of moment generating function and martingale methods, we study some properties of genralized random tree, including degree distribution, branching structure, degree of given vertices, maximum degree,etc.
The paper contains seven chapters:In the first chapter, we mainly introduce the history of the topic we will discuss and the work we have done; In the second chapter, we mainly study an invariant formula and the equivalence of two kinds of norms induced by generator of exclusion process and its pregenerator; In the third chapter, we give an upper bound of occupation time variance by use of the invariant formula in chapter 2; In the fourth chapter, we prove the existence and uniqueness of the martingale problem corresponding to the generator of exclusion on graph; In the fifth chapter, we discuss the problem of the number of vertices of subtree when we delete a node randomly; In the sixth chapter, we generalize the random uniform recursive tree and preferential attachment tree and study the degree distribution,
本文主要分成以下七个章节.在第一章,我们主要介绍了所研究问题的历史,现状和本文所做的工作；在第二章中,我们研究了一个变分公式和两类由生成元和准生成元所诱导出的范数的等价关系；在第三章中,利用第二章中的变分公式,我们给出了排它过程的占位时函数的方差的一个上界；第四章,我们主要讨论了图上排它过程的生成元对应的鞅问题的解的存在唯一性；在第五章,我们主要讨论了对于均匀递归树,随机删除一个顶点后,剩余子树的顶点数问题；在第六章中,我们对于均匀递归树和有偏好加点树进行了推广,并研究了在推广的随机树中的度分布,分枝结构和最大度问题.在第七章,我们继续讨论随机递归树,我们考虑一类树,其生成是在偶数时间,有偏好加点,在奇数时间均匀加点或以m为时间间隔,在m的倍数时间我们有偏好加点,在其它时间均匀加点,对于这样的树过程,我们考虑它们的度分布问题.
In this paper, we study the exclusion process and random tree by the methods of dualty, martingale, moment generating function, etc.
Firstly, we study the upper bound of the occupation time variance by some knowledge of random walk and Fourier transform; Secondly, we prove the existence of exclusion process on random graph, we transform the problem into the solution of the martingale problem corresponding to the generator of the exclusion, so the problem of existenc is settled; Thirdly, we study the problem of number of vertices in nodes deletion tree process by the means of moment generating function; Finally, by the means of moment generating function and martingale methods, we study some properties of genralized random tree, including degree distribution, branching structure, degree of given vertices, maximum degree,etc.
The paper contains seven chapters:In the first chapter, we mainly introduce the history of the topic we will discuss and the work we have done; In the second chapter, we mainly study an invariant formula and the equivalence of two kinds of norms induced by generator of exclusion process and its pregenerator; In the third chapter, we give an upper bound of occupation time variance by use of the invariant formula in chapter 2; In the fourth chapter, we prove the existence and uniqueness of the martingale problem corresponding to the generator of exclusion on graph; In the fifth chapter, we discuss the problem of the number of vertices of subtree when we delete a node randomly; In the sixth chapter, we generalize the random uniform recursive tree and preferential attachment tree and study the degree distribution,
引文
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[3]A.L.Barabasi,R.Albert,H.Jeong(1999)Scale-free characteristics of random networks: topology of the world wide web Physica A 281,69-77.
[4]A.Panholzer and H.Prodinger(2005) The level of nodes in increasing trees revisited. Ran-dom Struct. Algorithms
[5]A.Panholzer(2003) Non-crossing tree revisited:cutting down the spanning subtrees in discrete random walks. C.Banderier and C.Krattenthaler(eds),Discrete Mathmetics and Theoretical Science. Proceedings AC,265-276.
[6]Barabasi A.L., Albert R.and Jeong H.(1999) Mean-field theory for scale-free random networks.Physica A.272,173-187.
[7]B.Barabasi and R.Albert(1999) Emergence of scaling in random networks, Science 286, 509-512.
[8]Bala Bollobas(2001) Random Graphs. Combridge University Press. Combrige England.
[9]BelaBollobas and Oliver Riordan(2002) The diameter of a scale-free random graph, on intern
[10]B.Bollobas, O.riodan, J.Spencer and G.Tusnady(2001) The degree sequence of a scale-free random graph process, Random strucure and algorithms 18,270-290.
[11]B.Derride,B.Doucot and P.E.Roche(2003) Current fluctuations in the one dimensional symmetric exclusion process with open boundaries.preprint.
[12]B.Derrida,C.Enaud,C.Landim and S.Olla(2006) Fluctuations in the weakly asymmetric exclusion process with open boundary conditions.pre-print.
[13]Bllen Sadda(1987) A limit theorem for the position of a tagged particle in a simple exclusion process. Ann.Probab.13,375-381.
[14]Cedric Bernadin(2004) Fluctuations in the occupation time of a site in the asymmetric simple exclusion process.Ann.Probab.32,855-897.
[15]C.Cooper,A.Frieze(2003) A general model for web graphs, Random Struct.Algorithms 22, 33-335.
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[17]C.Landim,S.Olla and S.R.S.Varadhan(2000) Asymptotic behavior of a tagged particle in simple exclusion process. Bol.Soc.Bras.Mat.31,241-275.
[18]C.Landim, S.Olla and S.R.S.Varadhan(2002) Finite-dimensional approximation of the self-diffusion coefficient for the exclusion process. Ann.probab.30,483-508.
[19]C.Landim(1992) Occupation time large deviation for the symmetric simple exclusion process. Ann.Probab.20,206-231.
[20]C.Landim, H.T.Yau(1997) Fluctuation-dissipation equation of asymmetric simple exclu-sion processes.Probab.Theory Relat.Fields 108,321-356.
[21]C.Landim(1992) Conservation of local equilibrium for attractive particle system on Zd. Ann.Probab.21,1871-1902.
[22]C.Landim, S.Olla and H.T.Yau(1996) Some properties of the diffusion coefficient for asymmetric simple exclusion processes.Ann.Probab.24,1779-1808.
[23]C.Landim, S.Olla, S.R.S.Varadhan(2001) Symmetric simple exclusion process:regularity of the self-diffusion coefficient.Commun.Math.Ph-ys.224,307-321.
[24]C.Landim, A.Milanes, S.Olla(2006) Stationary and nonequilibrium fluctuations in bound-ary driven exclusion processes.preprint.
[25]Clifford, P. and Sudbury, A.(1973) A model for spatial conflict. Biometrika,60,581-588.
[26]C.Pigorsch and G.M.Schutz(2000) Shocks in the asymmetric simple exclusion process in a discrete-time update.J.Phys.A:Math.Gen.33,7919-7993.
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[29]Chun Su,Qunqiang,Feng and Zhishui Hu(2006) Uniform recursive trees:Branch-ing struc-ture and simple random downward walk.J.Math.An-al.Appl.315,225-243.
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[31]D.Han(1995) Uniqueness of infinite dimensional reaction diffusion process with multi-species. Chinese Journal of Contemporary Mathematics vol.16 No.4.
[32]D.Han, X.S.Zhang, W. A.Zheng Subcritical and supercritical size distributions in random coagulation-fragmentation process. Acta. Math.English series(to appear)
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