2022 Fiscal Year Final Research Report
Development of stochastic differential geometry associated with sub-Laplacians
| Project/Area Number |
18K03336
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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 | Kyushu University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | マリアバン解析 / サブラプラシアン / グルーシン作用素 / 熱核 |
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| Outline of Final Research Achievements |
Diffusion processes associated with sub-Laplacians of sub-Riemannian manifolds are constructed and several quantities appearing in their short time asymptotic behavior are computed concretely. In particular, the short time asymptotic behavior of heat kernel associated with the Grushin operator with general parameter a of real number, which is no longer equi-regular and give as the sum of the n-dimensional Laplacian and the m-dimensional Laplacian multiplied by the n-dimensional Euclidean norm to 2a-th power) is investigated in detail in the cases of on-diagonal and off-diagonal. In both cases, the dependence on the parameter a of quantities resulting in the asymptotic behavior is clarified. On the way, new theoretical results are obtained; among them are (i) the construction of diffusion processes on vector bundles acted by orthogonal groups, (ii)the reconstruction of the Malliavin calculus on manifolds, and (iii) the reformulation of the partial Malliavin calculus on Euclidean spaces.
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| Free Research Field |
確率解析
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| Academic Significance and Societal Importance of the Research Achievements |
多様体上の拡散過程の研究は，非退化微分作用素やヘルマンダー型と呼ばれるベクトル場の2乗和となる作用素で生成されるものが中心であり，サブリーマン多様体上のサブラプラシアンのように退化しさらにヘルマンダー型ではない生成作用素に対する確率解析的研究は非常に新しいものである．さらに，パラメータが自然数のグルーシン作用素については解析的研究が進んでいたが，パラメータが一般の正実数の場合の研究はなされていなかった．一般の場合の漸近挙動解析は，確率解析的手法による新たな成果である．さらにWatanabeの超関数理論に基づく偏マリアバン解析の体系化もまた確率解析にとって重要な理論的貢献である．
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