2023 Fiscal Year Final Research Report
On global properties and its stability of the paths of Markov processes and their additive functionals
| Project/Area Number |
19K03552
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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| Research Institution | Kansai University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
富崎 松代 奈良女子大学, その他部局等, 名誉教授 (50093977)
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | ディリクレ形式 / 飛躍拡散過程 / 非局所作用素 |
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| Outline of Final Research Achievements |
In our research, the Dirichlet form technique have been widely used to study the path properties of Markov processes we are dealing with. One of our main results is that, under appropriate conditions on the Levy densities for which the densities are allowed to degenerate or to diverge at 0, the jump-diffusion processes are constructed by using the Dirichlet form theory. Moreover the polarity of 0 is also investigated.
We have also consider the homogenization of the jump-diffusion processes via a 2-scale convergence method. In particular, the method is firstly used for the jump processes in our research, which have been considered in the diffusion processes case so far. Because of the method, we have had to assume that the diffusion coefficients are continuous, but the unfolding method, instead, will handle this restriction and the result could be relaxed to the case when the diffusion processes having drift term.
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| Free Research Field |
確率過程論
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| Academic Significance and Societal Importance of the Research Achievements |
飛躍をもつマルコフ過程論の研究は，マルコフ過程論の研究そのものに対する理論的重要性もさることながら，領域に穴や隙間があり，それらの領域へ粒子が流れ込まないための条件や，逆にそこへ集約させるようにするためにどのような制御が必要かという現実的な観点から見ても重要だと思われる． また，均質化法は，偏微分方程式論や確率論だけにとどまらず，マクロな法則からミクロな法則へ変化するときの極限状況を知る上でも非常に有効な理論であり，確率過程論の収束と関連付けることで，より具体的な変化を理解することができる．
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