2020/07/02 by Frederic Weber, Weber, Frederic, Rico Zacher +1
Mathematics · #39A12 (secondary) #47D07 #60J27 (primary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2007.01264
openalex publication_date 2020/07/02 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We consider continuous-time (not necessarily finite) Markov chains on\ndiscrete spaces and identify a curvature-dimension inequality, the condition\nCD_\Υ(\κ,\∞), which serves as a natural analogue of the\nclassical Bakry- 'Emery condition CD(\κ,\∞) in several respects. In\nparticular, it is tailor-made to the classical approach of proofing the\nmodified logarithmic Sobolev inequality via computing and estimating the second\ntime derivative of the entropy along the heat flow generated by the generator\nof the Markov chain. We prove that curvature bounds in the sense of\nCD_\Υ are preserved under tensorization, discuss links to other notions\nof discrete curvature and consider a variety of examples including complete\ngraphs, the hypercube and birth-death processes. We further consider power type\nentropies and determine, in the same spirit, a natural CD condition which leads\nto Beckner inequalities. The CD_\Υ condition is also shown to be\ncompatible with the diffusive setting, in the sense that corresponding hybrid\nprocesses enjoy a tensorization property.\n