2005 Fiscal Year Final Research Report Summary
Basic Research for a Singular Class of Measure-Valued Stochastic Processes and Associated Nonlinear Systems.
| Project/Area Number |
14540101
|
|---|
| 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 | SAITAMA UNIVERSITY |
|---|
Principal Investigator |
DOKU Isamu SAITAMA UNIVERSITY, Faculty of Education, Professor, 教育学部, 教授 (60207686)
|
|---|
| Project Period (FY) |
2002 – 2005
|
| Keywords | Measure-valued stochastic processes / Branching processes / Random measures / Interactive particle systems / Immigration superprocesses / Scaling / Limit theorem / Convergence problem |
|---|
| Research Abstract |
Our main interest is a class of singular superprocesses. First of all we proved that there exists a unique measure-valued branching Markov process for a wider class of singular branching rates, which includes a Fleischmann-Muller superprocess with hyperbolic branching rate. We also showed the regularity of their paths. Then we derived a sufficient condition for such a class of superprocesses to possess the finite time extinction property. Moreover, we extended those results to a class of generalized superprocesses with diffusion-like spatial motion.
Next we tried to obtain a generalization of Dawson-Li-Zhon convergence result to SCSM. Actually we considered a more complicated class of superprocesses with immigration, and proved a limit theorem that rescaled immigration superprocesses converges to a SCSM in the sense of distribution on the space of measure-valued continuous paths. Now we are planning a new research project that we aim at deriving a new type of limit theorem, where a superprocess more generalized than SCSM arises as a rescaled limit.
Furthermore, we proved an existence and uniqueness theorem for weak solution to various kinds of stochastic equations associated with such a class of singular superprocesses. For a specific example, we showed absolute continuity of the law of solution by applying stochastic analysis.
|
