Malliavin calculus for stochastic flows

某些无限粒子系统的相互作用是由无限维度空间上的随机差异方程（SDE's）定义的。这将是一个有趣的问题，要研究与那些SDE's相关的无限尺寸发生器的部分异常现象。目前研究的主要工作是对部分超弹性度和由SDE定义的无限维度空间的随机流动的研究，让a_0（x， x^^-）是R^N值的平滑函数， ν（dθ）是离散空间θ上的有限度量， β_t =（β^θ_t）是Wiener过程，让μ（d）是测量饱和度，使A_0（x， x ^^-）是R ^N值的平滑函数。考虑特定类型的SDE系统：对于u μ R^d，dx^u（t）=（μ a_0（x^u（t）， x^β（t））μ（d））dt+ μ ν（dθ）a_θ（x^u（t））οdβ^θ_t.假设x^u（0）在u和过程x（t）=（x^u（t）在希尔伯特空间H=（L^2（R^d， β（R^d），μ））^N。Let\： H ￣ R^M be a bounded linear mapping。随机变量π（x（T））定律的平滑密度的存在被称为SDE的部分高密度。介绍H：A_0= μ μ（du）μ（d）a_0（x^u， x^）·μ/（μ x^u），A_θ= μ（du）a_θ（x^u）·μ/（μ x^u）·partial Hormander theorem，其中在partial Hormander condition下的partial hypoellipticity holds为SDE's进行了Malliavin计算。部分霍尔曼德理论可以应用于由SDE's的特定系统定义的随机流的确定性的传播的问题。

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.