1999 Fiscal Year Final Research Report Summary
Perturbation of domain of diffusion processes with boundary conditions and its application to the boundary value problem
| Project/Area Number |
10640112
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
|
|---|
| Research Institution | Kanazawa University |
|---|
Principal Investigator |
TSUCHIYA Masaaki Kanazawa Univ. Faculty of Engineering ; Prof., 工学部, 教授 (50016101)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KAWAKAMI Hajime Akita Univ. Fac. Engi. Resource Sci. ; Asso. Prof., 工学資源学部, 助教授 (20240781)
KANJIN Yuichi Kanazawa Univ. Grad. School Nat. Sci. Tech. ; Prof., 大学院・自然科学研究科, 教授 (50091674)
OGAWA Shigeyoshi Kanazawa Univ. Faculty of Engineering ; Prof., 工学部, 教授 (80101137)
|
|---|
| Project Period (FY) |
1998 – 1999
|
| Keywords | diffusion process / diffusion equation / fundamental solution / superpositon of diffusion process / Hardy-type inequality / Hankel transform / noncausal stochastic equation / numerical method for solving |
|---|
| Research Abstract |
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.
|
