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Research on properties of Markov processes governed by the pseudo-differential operators with variable orders and application of the m to nonlinear analysis

变阶伪微分算子控制的马尔可夫过程性质研究及m在非线性分析中的应用

基本信息

批准号：
10640159
负责人：
NEGORO Akira
金额：
$ 0.9万
依托单位：
Shizuoka University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640159/
关键词：
Markov process transition density psudo-diffrential operator Le'vy測度

项目摘要

As is well known, under suitable conditions, it has been shown that there exist pure jump type Markov processes governed by Levy. generating operators with degenerate Levy mesures. So we would like to know what conditions these Markov processes have their transition densities under. Recently, by using MALLIAVIN calculus, Kunita has constructed transition densities of these Markov proceses in some class. So, we tried to adapt the pseudo-differential operators theory for this problem and restricted our study to the case that the supports of Levy measures degenerated into mutualy independent d lines for each x in RィイD1dィエD1. Cosequently, we have got that Markov processes governed the following generators, L have transition densities. The L is<<numerical formula>>where θィイD2jィエD2(x) (j = 1, 2,…, d) are smooth RィイD1dィエD1-valued functions with bounded derivatives on RィイD1dィエD1 and satisfy |θィイD2jィエD2(x)|=1(j = 1, 2,…, d). Putting Θ(x)=(θィイD21ィエD2(x), θィイD22ィエD2(x), …, θィイD2dィエD2(x)), we assume that the eigenvalues of Θ(x)*Θ(x) are unifomly bounded to the below. And also, α is a constant satisfying 1 < α < 2 and nィイD2jィエD2(x,y) (j = 1,…, d) are smooth funcutions with bounded derivatives satisfying usual coditions. Now, we are rounding off the above work. We regret to say that we were able to have no result about the relation between nolinear differential operators and stochastic processes. But while we were studing this problem, we had the following results.(1) A one dimensional hyperbolic equation uィイD2ttィエD2 - uィイD2xxィエD2 = 0 is treated under a free boundary condition uィイD32(/)XィエD3-uィイD32(/)tィエD3=QィイD12ィエD1. The existance and the uniqueness of a classical solution is established loccaly.(2) A weak solution to some forth order nonlinear parabolic equation is constructed by the method of time semidisceretization. A technique of geometoric measure theory is employed in order to obtain to obtain the convergence of the nonlinear terms.

众所周知，在适当的条件下，证明了存在由Levy控制的纯跳型马氏过程。用退化Levy测度生成算子。所以我们想知道，这些马尔可夫过程在什么条件下有它们的转移密度。最近，Kunita利用MALLIAVIN演算构造了这些马尔可夫过程在某类中的转移密度。因此，我们尝试采用伪微分算子理论来研究这个问题，并将我们的研究限制在Levy测度的支集退化为相互独立的d条线的情况下。同时，我们得到了马尔可夫过程控制下列生成元，L有转移密度。L是<<numerical formula>>，其中θ D2 j D2（x）（j = 1，2，...，d）是R D1 d D1-值光滑函数，在R D1 d D1上具有有界导数，满足|θイD2 j D2（x）|=1（j = 1，2，.，d）。设Θ（x）=（θ D21 D2（x），θ D22 D2（x），.，θ D2 d D2（x）），我们假设Θ（x）*Θ（x）的特征值一致有界于下。并且α是一个常数，满足1 < α < 2，n ∈ D2，j ∈ D2（x，y）（j = 1，...，d）是具有有界导数的光滑函数，满足通常的条件.现在，我们正在完成上述工作。我们遗憾地说，我们能够没有结果的非线性微分算子和随机过程之间的关系。但当我们研究这个问题时，我们得到了以下结果。(1)在自由边界条件uイD32（/）X D3-uイD32（/）t D3=QイD12 D1下处理一维双曲方程uイD2 tt D2-uイD2 xx D2 = 0。局部地证明了古典解的存在唯一性。(2)利用时间半离散化方法构造了一类四阶非线性抛物型方程的弱解。为了得到非线性项的收敛性，采用了几何测度理论的技巧。