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Curvature-dimension conditions under time change
Bang-Xian Han1, Karl-Theodor Sturm2
1School of Mathematical Sciences, University of Science and Technology of China, Hefei, China.
Annali Di Matematica Pura Ed Applicata
|March 7, 2022
Summary
This study details how synthetic lower Ricci bounds transform under time changes for both local Dirichlet forms and metric measure spaces. It provides precise formulas for the Bakry-Émery and Lott-Sturm-Villani curvature-dimension conditions.
Area of Science:
- Differential Geometry
- Analysis on Metric Spaces
- Probability Theory
Background:
- Ricci bounds are fundamental in geometry and analysis.
- Curvature-dimension conditions provide a way to define lower bounds on Ricci curvature.
- Time change is a transformation that alters the dynamics of a space.
Purpose of the Study:
- To derive precise transformation formulas for synthetic lower Ricci bounds under time change.
- To investigate the behavior of curvature-dimension conditions under time change for local Dirichlet forms.
- To analyze the transformation of curvature-dimension conditions in metric measure spaces under time change.
Main Methods:
- Derivation of transformation formulas for synthetic lower Ricci bounds.
- Analysis of the Bakry-Émery curvature-dimension condition for local Dirichlet forms.
- Study of the Lott-Sturm-Villani curvature-dimension condition for metric measure spaces.
Main Results:
- Precise transformation formulas for synthetic lower Ricci bounds are established.
- The study provides a clear understanding of how curvature-dimension conditions change under time transformations.
- The findings apply to both local Dirichlet forms and metric measure spaces.
Conclusions:
- The transformation formulas offer new tools for studying geometric properties under time change.
- This work bridges concepts from analysis on metric spaces and probability theory.
- The results are expected to have implications in geometric analysis and stochastic processes.
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