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

我们通过解析方法处理黎曼域上具有边界条件的扩散过程。对于倾斜反射边界条件的情况，通过构造相应扩散方程的边值问题的基本解，证明了该过程相对于黎曼体积的转移概率密度的存在性。构造是通过修改参数矩阵方法来完成的。它可以获得非齐次问题的解的完整积分表示以及域扰动下解的稳定性。然后，我们提供具有非整数阶平滑度的流形对的平滑结果。此外，我们还扩展了用二阶Ventsel边界条件构造扩散方程基本解的方法。接下来我们验证了扩散过程叠加的Feller性质。叠加与渗透模型有关。该问题简化为表明边界层上诱导积分微分方程存在强解。它是通过狄利克雷形式对格林核的一些考虑以及对边界层上导出的扩散方程的基本解的一些详细估计得出的。这是与 Yukio Ogura 和 Matsuyo Tomisaki 的合作成果。与该主题相关，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给出了求解此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，并得到了离散Hardy空间的刻画。

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.