2021 Fiscal Year Annual Research Report
| Project/Area Number |
21F50732
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| Allocation Type | Single-year Grants |
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| Research Institution | Shinshu University |
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Principal Investigator |
謝 賓 信州大学, 学術研究院理学系, 教授 (50510038)
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| Co-Investigator(Kenkyū-buntansha) |
ZHU MIN 信州大学, 理学部, 外国人特別研究員
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| Project Period (FY) |
2021-11-18 – 2024-03-31
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| Keywords | approximation / anisotropic / SPDEs / alpha-stable / Euler-Maruyama |
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| Outline of Annual Research Achievements |
We aim to study stochastic anisotropic partial differential equations and approximation of path-distribution dependent stochastic differential equations driven by alpha-stable noise.
(1) The first purpose is to study well-posedness for stochastic anisotropic partial differential equations with delay, which includes stochastic anisotropic p-Laplacian reaction-diffusion equations. The anisotropic exponent here means that the nonlinear operator is a sum of operators with different analytic and growth properties in all directions. In this half year, we mainly used our time to study relative references both in the theory of partial differential equations and stochastic analysis on infinite dimensional space. We also tried to weaken some conditions imposed in existing literatures. But we met some difficulties, which will be considered in the nest year.
(2) The second one is to show via interacting particle systems an approximation issue on a class of path-distribution dependent stochastic differential equations driven by alpha-stable noise. In this case, we founded that the well-known Zvonkin-type approach does not work for the case of SDEs with multiplicative noises. We tried another approach based on the estimate of the jump diffusion. We partially obtained a result on the convergence rate of Euler-Maruyama scheme under our framework, which should be improved.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
The study of the project is going a little slower than expected due to the influence of the COVID-19 pandemic. Our work involves primarily two parts as we mentioned above. To succeed in our study, it is important for us to attend conferences about SPDEs and numerical analysis on SDEs. It is also essential to discuss with specialists on these fields. Because of COVID-19 pandemic, we could not make progress as we expected.
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| Strategy for Future Research Activity |
Based on the current status, we will continue to study our general framework on stochastic anisotropic partial differential equations and try to study more important nonlinear examples. On the other hand, we will also study the well-posedness of path-distribution dependent SDEs driven by alpha-stable noise and the corresponding stochastic interacting particle systems. If possible, we will attend some relative conferences to collect the newest research results.
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