An anlytic approach to diffusion processes with second order Ventsel's boundary conditions and its applications

1.我们用二阶Ventsel边界条件处理扩散过程。通过构造相应扩散方程的基本解，证明了这类过程的转移概率密度的存在性。我们还证明了转移概率密度的严格正性。此外，利用转移概率密度，我们得到了区域边界上的局部时间势的显式公式，区域边界是扩散过程的状态空间。为了得到上述结果，我们研究了分数阶流形的C^∞光滑化和Holder映射空间上的Whitney拓扑。接下来，在不假定极限过程的基本测度是非退化的情况下，我们考虑了与局部型Dirichlet形式相关的马尔可夫过程的收敛。在这种情况下，极限过程一般不是扩散的，而近似过程是扩散的。因此，我们通过得到相应的带边界条件的积分-微分方程组，给出了极限过程的一个解析特征。这是与Y.Ogura和M.Tomisaki.4的合作成果。最后，利用罚函数法讨论了具有反射边界条件的随机微分方程解的强逼近问题。结合实解析方法，我们得到了关于Paley等式和Hausdorff算子的结果。

1. We treat diffusion processes with second order Ventsel's boundary conditions. The existence of a transition probability desity for such a process is verified ; it is done by constructing a fundamental solution of the corresponding diffusion equation. We also show the strict positivity of the transition probability density. Furthermore, using the transition probability density, we obtained an explicit formula for the potential of the local time on the boundary of the domain which is the state space of the diffusion process.2. To get the result mentioned above, we study C^∞ smoothing of manifolds with fractional order and the Whitney topology on the spaces of Holder maps.3. Next we consider the convergence of Markov processes associated with local type Dirichlet forms without assuming that the basic measure of the limit process is non- degenerate. In this case, the limit process is not diffusion in general, whereas the approximate processes are diffusion. Hence we give an analytic chacterization for the limit process by obtaining the corresponding integro-differential equation with boundary condition. This is a joint work with Y. Ogura and M. Tomisaki.4. Finally, using penalty method, strong aproximation to the solutions of stochastic differential equations with reflecting boundary condition is considered. In connection with real analytic approach, we obtain results on Paley's iequality and Hausdorff operator.