Corollary 11.23.17`21368BD09EBB`

Corollary 11.23.17.21368BD09EBB For these paths, the initial derivative of every $C^2$ convex entropy of each probability relative to $\nu _ K$ is nonpositive, while the total-variation distance between the two probabilities has positive initial derivative for sufficiently small $\varepsilon >0$.

Proof. The initial velocities are those of the logarithmic flow, and both initial forms are closed. The chain-rule calculation in the proof of proposition 11.23.8 therefore applies at $t=0$: its background term vanishes and its gradient term is nonpositive. The preceding proposition gives the assertion about the pair. $\square$

Cite this page

BibTeX citation

@misc{pome-tag-21368BD09EBB,
  author = {{Pome contributors}},
  title = {Corollary 11.23.17},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/21368BD09EBB/}
}

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHub Post a comment on GitHub

Corollary 11.23.1721368BD09EBB

Comments

Corollary 11.23.17`21368BD09EBB`