可加稳定过程的重点

本文研究了可加稳定过程的重点问题.利用自相交局部时计算可加稳定过程多重时的Hausdorff维数,证明了在适当条件下可加稳定过程重点的存在性.     

分数噪声驱动的随机热方程解的局部时

该文研究了可加分数噪声驱动的随机热方程解的碰撞局部时和相交局部时.运用局部非确定性和混沌分解等方法得到它们的存在性和光滑性.     

广义α-stable过程自相交局部时增量的Holder律及重点

文研究了 N 指标 d 维广义α-stable过程自相交局部时增量的Holder律. 并利用所得自相交局部时的性质, 证明了该过程重点的存在性, 得到了该过程多重时的Hausdorff维数及测度的下界.所得结论包含并推广了广义布朗单及stable单相应的结果.     

广义α-stable过程自相交局部时增量的Hlder律及重点

该文研究了N指标d维广义α-stable过程自相交局部时增量的Holder律．并利用所得自相交局部时的性质,证明了该过程重点的存在性,得到了该过程多重时的Hausdorff维数及测度的下界．所得结论包含并推广了广义布朗单及stable单相应的结果．     

下临界维数情形Stable过程投影相交局部时的上尾估计

利用其中一个稳定过程将Rd上的ρ个α稳定过程相交局部时测度投影到其相交点集合上的这种方式,在相交集合点上定义一个自然测度l.在p(d-α)＜d且d≥2的条件下,证明了对任意的有界集U,事件l(U)＞a的概率随着a的增长将以指数速度衰减.     

双分式Brown运动的自相交局部时和相遇局部时

考虑一个双分式Brown运动的局部时、自相交局部时和两个独立的双分式Brown运动的相遇局部时问题．通过双分式Brown运动的强局部不确定性、L^2收敛和混沌展开,验证自相交局部时和相遇局部时的存在性和光滑性．     

分式Brownian运动的多重相交局部时

本文研究了分式布朗运动的多重相交局部时的问题.利用白噪声分析的方法,获得了分式布朗运动的多重相交局部时的展开式.进行适当的截取,展开式在白噪声广义泛函意义下存在,并给出它们的核函数.推广了布朗运动的多重相交局部时.     

对称Stable过程的多重自相交局部时研究

设X_t是取值于R²中阶为β的对称stable过程.本文主要研究X_t的k-重自相交局部时αk (x;t)及其重整化自相交局部时αk (0;t),其中k≥2.首先,当x≠0时,对所有p> 0,考虑αk(x;t)在L^p(Ω)中的存在性.其次,分别给出α_k(x;t)关于时间变量t和空间变量x的H?lder连续性条件.最后,由于α_k(0;t)不存在,故考虑其重整化局部时?_k(0;t)在L²(Ω)中的存在性问题.     

超凸空间中的选择定理及弱外超凸集的性质

本文利用半序方法,研究具有外超凸集值的带次可加模的集值映象的保不动点集,带相同次可加模的单值选择,获得了这类单值选择的存在性及唯一性条件.且研究了弱外超凸集的一些相交性质.     

d-维布朗运动局部时的刻画及可加泛函的表示定理

在本文里,我们定义了高维布朗运动的面局部时和有界区域的边界局部时,并用 Dirichlet 形式与随机分析理论证明布朗运动的面局部时对应的光滑测度正好是超平面上的面测度.作为上述结果的应用,我们还得到高维布朗运动可加泛函关于局部时的表示定理.     

Regularity of local times of random fields

In this paper, we study the fractional smoothness of local times of general processes starting from the occupation time formula, and obtain the quasi-sure existence of local times in the sense of the Malliavin calculus. This general result is then applied to the local times of N-parameter d-dimensional Brownian motions, fractional Brownian motions and the self-intersection local time of the 2-dimensional Brownian motion, as well as smooth semimartingales.     

Self-intersection Local Time of Planar Brownian Motion Based on a Strong Approximation by Random Walks

The main purpose of this work is to define planar self-intersection local time by an alternative approach which is based on an almost sure pathwise approximation of planar Brownian motion by simple, symmetric random walks. As a result, Brownian self-intersection local time is obtained as an almost sure limit of local averages of simple random walk self-intersection local times. An important tool is a discrete version of the Tanaka?CRosen?CYor formula; the continuous version of the formula is obtained as an almost sure limit of the discrete version. The author hopes that this approach to self-intersection local time is more transparent and elementary than other existing ones.     

Bivariate occupation measure dimension of multidimensional processes

Bivariate occupation measure dimension is a new dimension for multidimensional random processes. This dimension is given by the asymptotic behavior of its bivariate occupation measure. Firstly, we compare this dimension with the Hausdorff dimension. Secondly, we study relations between these dimensions and the existence of local time or self-intersection local time of the process. Finally, we compute the local correlation dimension of multidimensional Gaussian and stable processes with local Hölder properties and show it has the same value that the Hausdorff dimension of its image have. By the way, we give a new a.s. convergence of the bivariate occupation measure of a multidimensional fractional Brownian or particular stable motion (and thus of a spatial Brownian or Lévy stable motion).     

Large deviations estimates for self-intersection local times for simple random walk in {\mathbb{Z}}^3

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.      

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness (in the sense of Meyer- Watanabe) of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.     

Clark Representation for Local Times of Self-Intersection of Gaussian Integrators

Ukrainian Mathematical Journal - We prove the existence of multiple local times of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernels,,...     

一类S′(R1)值OU过程的自交局部时

Gauss型S′(R~d)值OU过程可由超过程在一定条件下取波动极限得到。本文讨论了一类新的S′(R~1)值OU过程的自交局部时的存在性,连续性。证明了这些性质与粒子的分枝强度和交互作用参变量的依赖关系。还讨论了当自交局部时不存在时,经过重整化的近似自交局部时满足一定的收敛性质。     

Intersection Local Times as Generalized White Noise Functionals

For any dimension we present the expansions of Brownian motion self-intersection local times in terms of multiple Wiener integrals. Suitably subtracted, they exist in the sense of generalized white noise functionals; their kernel functions are given in closed (and remarkably simple) form.     

Hilbert-valued self-intersection local times for planar Brownian motion

Dynkin's construction for self-intersection local time of a planar Wiener process is extended to Hilbert-valued weights. In Dynkin's construction, the weight is bounded and measurable. Since the weight function describes the properties of the medium in which the Brownian motion moves, relative to the external medium's properties, the weight function can be random and unbounded. In this article, we discuss the possibility to consider Hilbert-space-valued weights. It appears that the existence of Hilbert-valued Dynkin-renormalized self-intersection local time is equivalent to the embedding of the values of Hilbert-valued weight into a Hilbert–Schmidt brick. Using Dorogovtsev's sufficient condition for the embedding of compact sets into a Hilbert–Schmidt brick in terms of an isonormal process, we prove the existence of Hilbert-valued Dynkin-renormalized self-intersection local time. Also using Dynkin's construction we construct the self-intersection local time for the deterministic image of the planar Wiener process.