Perturbation of domain of diffusion processes with boundary conditions and its application to the boundary value problem

本文用解析方法处理黎曼域上带边界条件的扩散过程。对于斜反射边界条件的情况，通过构造相应扩散方程边值问题的基本解，证明了该过程关于Riemannian体积的转移概率密度的存在性。构造是通过修改参数化方法完成的。它产生获得一个完整的积分表示的非齐次问题的解决方案和稳定的解决方案下的扰动的区域。然后给出了光滑阶为非整数的流形对的光滑性的一个结果。进一步将构造基本解的方法推广到二阶Ventsel边界条件下的扩散方程，并证明了扩散过程叠加的Feller性质。这种叠加与磁导模型有关。问题归结为边界层上诱导积分微分方程强解的存在性。它遵循一些考虑绿色内核通过Dirichlet形式和一些详细的估计基本解的扩散方程上的边界层。这是与小仓幸雄和富崎松代的合作。与此相关的主题，我们也考虑非因果或非线性随机方程。从随机数值分析的角度给出了求解这类随机方程的数值方法，最后结合真实的分析方法，给出了Hankel空间的Hardy型不等式，并得到了离散哈代空间的一个特征.

We treat diffusion processes with boundary conditions on Riemannian domains by analytic method. For the case of the oblique reflecting boundary condition, the existence of the transition probability density with respect to the Riemannian volume for such a process is proved by constructing a fundamental solution to the boundary value problem for the corresponding diffusion equation. The construction is done by modifying the parametrix method. It yields obtaining a complete integral representation for solutions to the nonhomogeneous problem and the stability of the solutions under the perturbation of the domain. Then we provide a result on smoothing for manifold pairs with non-integral order of smoothness. Furtheremore, we extend the method of constructing fundamental solutions to diffusion equations with second-order Ventsel's boundary conditions.Next we verify the Feller property of the superposition of diffusion processes. The superposition is related to permeance model. The problem is reduced to showing the existence of a strong solution to the induced integro-differential equation on the boundary layer. It follows from some consideration on the Green kernel through the Dirichlet form and some detailed estimates of the fundamental solution to the diffusion equation induced on the boundary layer. This is a joint work with Yukio Ogura and Matsuyo Tomisaki.Related to the subject, we also consider noncausal or nonlinear stochastic equations. Some numerical method for solving such stochastic equations is given in a point of view of the stochastic numerical analysis.Finally, in connection with real analytic approach, we present Hardy-type inequalities for Hankel transfoms and obtain a characterization of discrete Hardy spaces.