Open AccessThe Łojasiewicz inequality for free energy functionals on a graph
Author(s) -
Kongzhi Li,
Xiaoping Xue
Publication year - 2022
Publication title -
communications on pure and applied analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.077
H-Index - 42
eISSN - 1553-5258
pISSN - 1534-0392
DOI - 10.3934/cpaa.2022066
Subject(s) - mathematics , exponent , bounded function , riemann hypothesis , graph , corollary , combinatorics , exponential function , discrete mathematics , mathematical analysis , pure mathematics , philosophy , linguistics
Rencently Chow, Huang, Li and Zhou proposed discrete forms of the Fokker-Planck equations on a finite graph. As a primary step, they constructed Riemann metrics on the graph by endowing it with some kinds of weight. In this paper, we reveal the relation between these Riemann metrics and the Euclidean metric, by showing that they are locally equivalent. Moreover, various Riemann metrics have this property provided the corresponding weight satisfies a bounded condition. Based on this, we prove that the two-side Łojasiewicz inequality holds near the Gibbs distribution with Łojasiewicz exponent \begin{document}$ \frac{1}{2} $\end{document} . Then we use it to prove the solution of the discrete Fokker-Planck equation converges to the Gibbs distribution with exponential rate. As a corollary of Łojasiewicz inequality, we show that the two-side Talagrand-type inequality holds under different Riemann metrics.