the regularity of stochastic flows on functional spaces

具有时滞系数的随机微分方程组是具有泛函系数的随机微分方程的典型例子。这些随机微分方程在金融数学理论中经常出现。Malliavin微积分将有效地分析这些SDE。研究人员之一A.Takeuchi应用Malliavin微积分研究了具有函数系数的SDE。他证明了在一定的非退化条件下，由Levy过程驱动的随机微分方程解的边际分布具有Radon-Nikodym密度。勒贝格措施。在Hormander条件下，证明了具时滞系数的随机微分方程解的边缘分布在Hormander条件下是光滑的。变阶抛物型拟微分算子L=∂t-p(x，Dx)的鞅问题由首席研究员T.Komatsu研究。假设函数a(X)是光滑的，但符号p(x，e)在x中并不总是可微的。在非退化条件下，他证明了这类鞅问题解的唯一性。他的研究的要点是得到与鞅问题的解相关的预解算子的Lp估计。他利用伪微分算子理论和奇异积分的广义Calderon-Zygmund不等式得到了这一结果。作为研究的结果，构造了生成元为L的马尔可夫过程，并对其进行了刻画。马尔可夫过程可以称为具有扰动的类稳定过程，研究人员在研究对象范围内对各种问题进行了研究。M·吉田研究了Denavy动力系统及其维群。M.Nishio研究了α-抛物Bergman空间上的测度和算子。这项研究的一些结果已经发表在数学期刊上。

Stochastic differential equations (SDE's) with time delay coefficients are typical examples of SDE's with functional coefficients. These SDE's appears frequently in the mathematical theory on the finance. The Malliavin calculus would be effective to analyze these SDE's. A. Takeuchi, one of investigators, studied SDE's with functional coefficients applying the Malliavin calculus. He proved that, under a certain non-degenerate condition, marginal distributions of solutions to SDE's driven by Levy processes have Radon-Nikodym densities w.r.t. the Lebesgue measure. He also proved that the marginal distributions of solutions to SDE's with time delay coefficients, driven by Brownian motions, have smooth densities under the Hormander condition.Martingale problems for parabolic pseudo-differential operators L = ∂t - p(x, Dx) of variable order a(x) < 2 are studied by T. Komatsu, the head investigator. The function a(x) is assumed to be smooth, but the symbol p(x, e) is not always differentiable in x. He proved the uniqueness of solutions to the martingale problems under a non-degenerate condition. The essential point in his study is to obtain the LP-estimate for resolvent operators associated with solutions to the martingale problem. He obtained that by making use of the theory of pseudo-differential operators and a generalized Calderon - Zygmund inequality for singular integrals. As a consequence of the study, the Markov process with the generator L is constructed and characterized. The Markov process may be called a stable-like process with perturbations.Investigators of this resurch studied various problems going over the extent of the research subject. M. Yoshida studied on the Denjoy dynamical sytstems and their dimension groups. M. Nishio studied measures and operators on α-parabolic Bergman spaces. Some of results of the resurch have published on mathematical journals.