Malliavin calculus for stochastic flows

随机流的 Malliavin 微积分

基本信息

批准号：
15540133
负责人：
KOMATSU Takashi
金额：
$ 2.11万
依托单位：
Osaka City University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

项目摘要

Certain infinite particle systems with interactions are defined by stochastic differential equations (SDE's) on infinite dimensional spaces. It would be an interesting problem to study on the partial hypoellipticity of infinitesimal generators on infinite dimensional spaces associated with those SDE's. The main work of the present research is the study on the partial hypoellipticity vi the Malliavin calculus for stochastic flows on infinite dimensional spaces defined by SDE's.Let a_0(x, x^^-) be an R^N-valued smooth function on R^NxR^N, ν(dθ) be a finite measure on a discrete space θ, β_t=(β^θ_t) be a Wiener process, and let μ(dυ) be a measure satisfying a strong integrability condition. Consider a system of SDE's of the specific type : for u∈R^d,dx^u(t)=(∫a_0(x^u(t), x^υ(t))μ(dυ))dt+∫ν(dθ)a_θ(x^u(t))οdβ^θ_t.Assume that x^u(0) is smooth in u and the process x(t)=(x^u(t)) takes its values in the Hilbert space H=(L^2(R^d, β(R^d),μ))^N. Let π: H→R^M be a bounded linear mapping. The existence of the smooth density of the law of the random variable π(x(T)) is called the partial hypoellipticity of the SDE. Introduce the partial Hormander condition for vector fields on H:A_0=∫∫μ(du)μ(dυ)a_0(x^u, x^υ)・∂/(∂x^u), A_θ=∫μ(du)a_θ(x^u)・∂/(∂x^u)・The partial Hormander theorem that the partial hypoellipticity holds under the partial Hormander condition is proved proceeding the Malliavin calculus for SDE's on the Hilbert space H. The partial Hormander theorem can be applied to the problem about the propagation of absolute continuity of measures induced by stochastic flows defined by a certain system of SDE's.

具有相互作用的某些无限粒子系统由无限尺寸空间上的随机微分方程（SDE）定义。研究与这些SDE相关的无限尺寸空间上无限发电机的局部低细胞低纤维化是一个有趣的问题。本研究的主要工作是关于局部低纤维化的研究，即在无限尺寸空间上进行随机流的malliavin conculus，由sde's.s.let a_0（x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x，x ^^ - ）是r^n v n v n v nxr^nxr^nxr^n，ν（d tibe）的平稳函数。 β_T=（β^θ_t）是一个维也纳过程，让μ（dυ）为满足强积分条件的测量值。考虑特定类型的SDE系统：对于U∈R^d，dx^u（t）=（∫A_0（X^u（t），X^U（t），X^υ（t））μ（dυ（dυ））dt+∫ν（dθ）a_θ（x^u（x^u（x^u（t）））other它在希尔伯特空间中的值h =（l^2（r^d，β（r^d），μ））^n。令π：h→r^m为有界的线性映射。随机变量π（x（t））定律的平滑密度的存在称为SDE的部分低纤维化。介绍H：A_0 =∫∫μ（dυ）μ（x^u，x^∂，X^∂）・/（∂x^u），a_θ=∫μ（du∫μ（du）a_θ（x^u）a_θ（x^u）a_0 =∫∫μ（dυ）a_0（x^u，x^∂）a_0（x^u，x^U）的部分hormander条件事实证明，霍尔曼德的条件是在希尔伯特空间H上的SDE的Malliavin演算。部分荷尔曼德定理可以应用于关于由由某个SDE系统定义的随机流引起的测量绝对连续性传播的问题。