the regularity of stochastic flows on functional spaces

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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.