IJPAM: Volume 46, No. 3 (2008)

A NOTE ON THE FISHER INFORMATION
METRIC AND HEAT KERNELS

Mitsuhiro Itoh$^1$, Hiroyasu Satoh$^2$, Yuichi Shishido$^3$
$^{1,2}$Institute of Mathematics
University of Tsukuba
Tsukuba, Ibaraki, 305-8571, JAPAN
$^1$e-mail: itohm@sakura.cc.tsukuba.ac.jp
$^2$e-mail: hiroyasu@math.tsukuba.ac.jp
$^3$Meikei High School
Inarimae, Tsukuba, Ibaraki, 305-8502, JAPAN
e-mail: shishido@meikei.ac.jp


Abstract.A heat kernel map is defined from a complete Riemannian manifold $(X,h)$ into the space ${\cal P}(X)$ of probabilities on $X$ whose density function is positive. By pulling back, by this heat kernel map, the Fisher information metric on it, we assert that this induced metric is homothetic to the original metric of $X$, provided $(X,h)$ is a rank one symmetric space of non-compact type and, also discuss monotonicity of the homothety constant in time.

Received: April 30, 2008

AMS Subject Classification: 53C35, 53C42, 58B20

Key Words and Phrases: Fisher information metric, heat kernel, rank one symmetric space of non-compact type

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 46
Issue: 3