随机环境中非紧邻随机游动及高维随机游动的极限理论

Attributing to the randomization of the environment, a lot of new phenomena of random walk arise. Since it was first studied, Random Walk in Random Environment (RWRE) has been one of the most active field in probability theory..In one-dimensional case, based on our former works, for RWRE with unbounded jumps, firstly we intend to characterize its recurrence/transience criterion and present a complete solution of its law of large numbers. Secondly, by combining the RWRE with unbounded jumps and the skeleton process of Birth and Death Process in Random Environment (BDPRE) with bounded jumps, we study the asymptotic theory of BDPRE with bounded jumps. Furthermore, considering the RWRE with small perturbation on the environment, by analyzing the traps formed by the environment, we give a criterion which determines the finiteness and infiniteness of the number of its cut points. Fourthly, by developing the large deviations principle of random matrices, we estimate accurately the slowdown properties of RWRE and BDPRE with bounded jumps. At last, we discuss the existence of the stationary distribution of RWRE and reveal the new phenomenon of ergodicity speed caused by the randomization of the environment..In higher dimensional case (d>1), there is a famous Sznitman Conjecture about the ballisticity of RWRE: under the condition of uniform ellipticity, once the RWRE is transient, it runs to infinity with positive speed. We plan to prove partially this conjecture, so that we can understand more deeply how the interaction of the dimension and the randomization of the environment will affect the speed of the walk. .Finally, this project aims to study the infinite-type branching process which is still a virgin field. The main concern is its criticality. By studying the infinite-type branching process, our purpose is to construct further the branching structure for random walk with unbounded jumps, developing new approaches for the study of RWRE with unbounded jumps.

随机环境赋予游动全新的性质，随机环境中随机游动（RWRE）自其诞生始，一直是概率论中最活跃的领域之一。.首先，一维情形，以申请人前期工作为基础，讨论无界跳幅RWRE的常返暂留性，全面解决大数定律；然后将无界跳幅RWRE与有界跳幅随机环境中生灭过程（BDPRE）的骨架过程有机结合，研究有界跳幅BDPRE的渐进理论；解析环境陷阱的约束力，研究带扰动RWRE分割点的有限性；发展随机矩阵大偏差工具，准确刻画有界跳幅RWRE及BDPRE的慢速度；讨论平稳分布的存在性，探索正常返情形随机环境给游动遍历速度带来的新现象。.其次，高维情形，拟部分推进Sznitman猜想：一致椭圆条件下，高维RWRE一旦暂留，其速度必定非零。进而深入认识环境及维数交互作用对游动速度的影响。.再次，无穷物种分枝过程仍是一个新领域，我们拟讨论其临界性，建立无界跳幅随机游动的分枝结构，为无界跳幅RWRE的研究提供新工具。

本课题主要围绕随机环境中随机游动（RWRE）、转移概率带扰动的随机游动（RW）而展开，其中也涉及到随机环境中多物种分枝过程的极限理论等内容。.首先，对无界跳幅RWRE，在环境满足一致椭圆、尾部概率多项式衰减的条件下，利用“从粒子看环境”的方法，证明了大数定律。无界跳幅RWRE与随机环境中有界跳幅生灭过程有密切联系，以无界跳幅RWRE的大数定律为基础，导出了随机环境中有界跳幅生灭过程的大数定律。. 第二，考虑转移概率带扰动的有界跳幅RW，这类RW通常被称为Lamperti RW。利用正矩阵乘积的遍历定理及连分数的极限理论，对（2,1）RW，给出了其分割点有限性的判定准则，对（1，2）RW，给出了其“掠过的点”有限性的判定准则。此外，利用RW的分枝结构为工具，研究了（2,1）RW及（2,1）生灭过程的遍历性并给出了平稳分布。对于Lamperti RW的情形，用矩阵乘积的遍历性，给出了平稳分布尾部概率准确的衰减速度。. 第三，RWRE的环境可能依赖于一些参数，从统计观点出发，基于游动的样本轨道，给出环境参数的估计是很重要的。对轨道进行分解，构造了(L,1)RWRE环境参数的极大似然估计（MLE）。利用RWRE的分枝结构可证明，MLE可表示为随机环境中带移民的多物种分枝过程的泛函。通过这种手段，证明了所构造的MLE的相合性。. 第四，考虑向右暂留的有界跳幅RWRE，由于是非紧邻的，故趋向正无穷的过程中，它可能掠过很多点。我们证明，游动掠过占正整数轴一个正比例的点。换句话说，游动的轨道只访问过正整数轴一定比例的点。. 最后，对于一类单边无界跳幅的RW，通过轨道分解，建立了其分枝结构。该分枝结构对应于一个无穷物种的分枝过程。于是游动的首中时可以表示为无穷物种分枝过程的泛函，以分枝过程为工具，可研究首中时的分布、期望等性质。. 课题研究成果一定程度上丰富和完善了RWRE的理论体系，提出了游动掠过的点等新的概念，相关研究内容有进一步拓展的空间。