the regularity of stochastic flows on functional spaces

带时滞系数的随机微分方程是带泛函系数的随机微分方程的典型例子。这些SDE在金融数学理论中经常出现。Malliavin演算可以有效地分析这些SDE。研究者之一A. Takeuchi运用Malliavin演算对SDE的泛函系数进行了研究。他证明了在一定的非简并条件下，由Levy过程驱动的SDE解的边际分布在勒贝格测度下具有Radon-Nikodym密度。他还证明了在布朗运动驱动下具有时滞系数的SDE方程解的边际分布在Hormander条件下具有光滑密度。研究了变量阶为a(x) < 2的抛物型伪微分算子L =∂t - p（x, Dx）的鞅问题。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。他证明了在非退化条件下鞅问题解的唯一性。他的研究重点是得到与鞅问题解相关的可解算子的lp估计。他利用伪微分算子理论和广义的奇异积分Calderon - Zygmund不等式得到了这个结果。作为研究的结果，构造了具有生成器L的马尔可夫过程并对其进行了表征。马尔可夫过程可称为具有扰动的类稳定过程。这项研究的研究人员研究了涉及研究课题范围的各种问题。M. Yoshida研究了Denjoy动力系统及其维群。M. Nishio研究了α-抛物型Bergman空间上的测度和算子。这项研究的一些结果已发表在数学期刊上。

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.