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

用解析方法处理黎曼域上具有边界条件的扩散过程。对于斜反射边界条件，通过构造相应扩散方程边值问题的基本解，证明了该过程相对于黎曼体积的转移概率密度的存在性。采用修改参数法进行构造。给出了非齐次问题解的完整积分表示及其在定义域摄动下解的稳定性。然后给出了非整阶光滑流形对的光滑性结果。在此基础上，进一步推广了具有二阶Ventsel边界条件的扩散方程基本解的构造方法。接下来我们验证扩散过程叠加的Feller性质。叠加与渗透率模型有关。该问题被简化为表明边界层上诱导积分-微分方程的强解的存在性。本文从狄利克雷形式对格林核的一些考虑和边界层扩散方程基本解的一些详细估计出发。这是小仓幸雄和富崎松代的合作作品。与本课题相关，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，结合实解析方法，给出了Hankel变换的Hardy型不等式，得到了离散Hardy空间的一个表征。

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.