Combinatorics Seminar 2018

Speaker: Fan Cheng (程帆), Shanghai Jiao Tong University

Date: Thu, Jan 10, 2019

Time: 16:20 - 17:00

Venue: Xiayuan 211

Title: On the complete monotonicity of heat equation


A function $f(t)$ is called complete monotone, iff all the signs of its derivatives are alternating in + and -; e.g., $1/t$ and $e^{-t}$. The complete monotone index (CMI) is generalized to denote the length of the alternating sequence of + and -. By this means, both the CMI of $1/t$ and $e^{-t}$ are $+\infty$. Gaussian complete monotonicity conjecutre (GCMC): let $f(x,t)$ be the solution to the heat equation on the line, then the entropy $h(f(x,t))$ is complete monotone in $t$. By means of information theory: Let $X$ be an arbitrary random variable which is indepdent of the standard normal distrubtion (Gaussian distribution) $Z \sim \mathcal{N}(0,1)$, then $h(X+\sqrt{t}Z)$ is complete monotone in $t$.

The conjeture was first studied by H. P. McKean in 1966. We rediscovered it in 2015. In this talk, we will introduce our conjecture and progress on the complete monotonicity conjecture on heat equation.

Slides: View slides