2025/04/02 by Li, Xiang-Dong, Zhang, Enrui · 1 citation
#53C21 #60H30 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Primary 53C23 #Probability (math.PR) #Secondary 60J60
paper · doi:10.48550/arxiv.2504.01864
In this paper, we prove the W-entropy formula and the monotonicity and rigidity theorem of the W-entropy for the heat flow on RCD(K, N) and RCD(K, n, N) spaces (X, d, μ), where K∈ ℝ, n∈ ℕ is the geometric dimension of (X, d, μ) and N≥ n. We also prove the K-concavity of the Shannon entropy power on RCD(K, N) spaces. As an application, we derive the Shannon entropy isoperimetric inequality and the Stam type logarithmic Sobolev inequality on RCD(0, N) spaces with maximal volume growth condition. Finally, we prove the rigidity theorem for the Stam type logarithmic Sobolev inequality with sharp constant on noncollapsing RCD(0, N) spaces.