Logarithmic Sobolev Inequalities

Abstract

There is an interesting connection between our considerations here and L. Gross’s theory of logarithmic Sobolev inequalities. For our purposes, it is best to describe a logarithmic Sobolev inequality in the following terms. Let {P_x: x ∈ E} satisfy (S.C.) with respect to m ∈ m₁ (E). A logarithmic Sobolev inequality is a statement of the form:

$$ {J_m} \leqslant \alpha {J_{\sigma }} $$

((9.1))

for some α > 0, where J_m: m₁(E) → [0, ∞) ∪ {∞} is defined by: Obviously, (9.1) has interesting implications for the large deviation theory associated with {P_x: x ∈ E}.