Multilateral research of stochastic analysis in infinite dimensional spaces

完成了对半群控制性和半群的缠绕性质的研究。半群支配表示|T^^→_tu|[小于或等于]T_t|u|对半群T^^→_t和T_t成立。这里假设u是一个向量值函数。典型的例子是黎曼流形上的微分形式。关于生成元的表征是已知的。本文给出了平方场算子的一个充分条件。关键的假设是积极性和局部性。缠绕性是如下性质：对于两个生成元L，L^→，认为DL=L^→D+R。我们用半群和预解的形式给出了充分必要条件。我们的目的是重建Bakry-Emery T_2理论，但我们已经成功地将扩散过程包含在边界条件中，它给出了Bakry-Emery T_2理论的推广.在应用方面，我们可以利用它们来证明Littlewood-Paley不等式和L^p乘子定理.实际上，我们已经考虑了带边界的黎曼流形上的布朗运动，并在边界的第二基本形式为正的假设下证明了它的Littlewood-Paley不等式。利用这一性质，我们可以证明Riesz变换的L^p有界性。L p乘子定理给出了φ和A上φ(A)是L p上有界算子的一个充分条件，Stein对对称扩散过程的生成元证明了这一点，其中φ是拉普拉斯变换型的。我们证明了同样的结果也适用于黎曼流形上的Hodge-Kodaira算子。

We have accomplished the research of the semigroup domination and the intertwining property of semigroups. The semigroup domination stands for that |T^^→_tu|【less than or equal】T_t|u| holds for semigroup T^^→_t and T_t. Here u are supposed to be a vector valued function. Typical example is a differential form on a Riemannian manifold. The characterization in terms of the generator is known. We give here a sufficient condition in terms of square field operator. Crucial assumptions are the positivity and the locality. Intertwining property is the following property: for two generator L, L^^→, it holds that DL = L^^→D + R. We give necessary and sufficient conditions in terms of the semigroup and the resolvents. Our aim has been to reconstruct the.Bakry-Emery T_2 theory but we have succeeded in including diffusion processes with boundary condition and it gives a generalization of Bakry-Emery T_2 theory.For application, we can make use of them to show the Littlewood-Paley inequality and the L^p multiplier theorem. In fact, we have considered the Brownian motion on an Riemannian manifold with boundary and show the Littlewood-Paley inequality for it under the assumption of positivity of the second fundamental form of the boundary. Making use of this, we can show the L^p boundedness of the Riesz transformation. L^p multiplier theorem is to give an sufficient condition on φ and A so that φ(A) is a bounded operator on L_p. Stein showed this for a generator of a symmetric diffusion process with φ being of Laplace transform type. We have shown that the same result holds for the Hodge-Kodaira operator on a Riemannian manifold.