Application of Optimal Transport and Information geometry to Metric Measure Spaces

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17536
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: TAKATSU Asuka 首都大学東京, 理学研究科, 准教授 (90623554)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: 最適輸送理論 / 情報幾何 / 測度距離空間 / 熱流 / エントロピー / 凸性

Outline of Final Research Achievements

The aim of this research is to combine Wasserstein geometry with Information geometry and application of the both geometry to geometric analysis by taking account of the fact that the asymptotic behavior of the heat equation plays an important role in geometric analysis. In the joint work with Kazuhiro Ishige (the university of Tokyo) and Paolo Salani (university of Florence), we proved that the logarithmic concavity is not the strongest concavity preserved by the heat equation. Moreover we clarified what is the strongest concavity preserved by the heat equation. From the nature of the strongest concavity, it is expected to generalize the theory of Wasserstein geometry and Information geometry together by using the strongest concavity. In this way, this research made some contributions to a combining Wasserstein geometry with Information geometry.

Free Research Field

幾何解析

Academic Significance and Societal Importance of the Research Achievements

H.J.BrascampとE.H.Liebが1976年に打ち出した不等式以降、対数凹性は熱流に最適とされ、熱流の漸近解析や形状解析などの大域解析の軸になっていた。よって上記で述べた熱流で保たれる最強の凹性が対数凹性ではないという事実、および熱流で保たれる最強の凹性を決定した事実は、熱流の大域解析に新機軸を与える。特に熱流におけるこの最強の凹性の保存則を導くエントロピーの凸性、およびそのエントロピーの凸性から得られる幾何学的条件を明らかにすることは、熱流の大域挙動に関わる幾何学、曲率の条件をかした幾何解析に新展開を持ち込むことが期待できる。