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

我们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，通过构造相应扩散方程边值问题的基本解，证明了该过程关于黎曼体的转移概率密度的存在性。构造是通过修改Parametrix方法完成的。得到了非齐次问题解的完整积分表示以及在区域摄动下解的稳定性。然后，我们给出了非整数阶流形对的光滑化结果。进一步，我们推广了构造二阶Ventsel边界条件下扩散方程基本解的方法，并证明了扩散过程叠加的Feller性质。这种叠加与渗透率模型有关。问题归结为边界层上诱导积分-微分方程解的存在性。通过对格林核的Dirichlet形式的讨论，以及对边界层上的扩散方程基本解的一些详细估计，得到了这一结果。这是与小仓幸男和松代智作的合作成果。与主题相关，我们还考虑了非因果或非线性随机方程。从随机数值分析的角度出发，给出了求解这类随机方程的一些数值方法。最后，结合实解析方法，给出了Hankel变换的Hardy型不等式，并得到了离散Hardy空间的一个刻画。

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.