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

1. 我们用二阶文塞尔边界条件来处理扩散过程。验证了该过程的转移概率密度的存在性；它是通过构造相应扩散方程的基本解来实现的。我们还证明了跃迁概率密度的严格正性。进一步，利用跃迁概率密度，我们得到了扩散过程状态空间域边界上局部时间势的显式表达式。为了得到上述结果，我们研究了分数阶流形的C^∞平滑和Holder映射空间上的Whitney拓扑。在不假设极限过程的基本测度是非退化的情况下，我们考虑了与局部狄利克雷形式相关的马尔可夫过程的收敛性。在这种情况下，极限过程不是一般的扩散，而近似过程是扩散。在此基础上，给出了极限过程的解析表征，得到了相应的带边界条件的积分-微分方程。这是与小仓先生和富崎先生合作的作品。最后，利用惩罚法研究了具有反射边界条件的随机微分方程解的强逼近性。结合实解析方法，得到了关于Paley不等式和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.