Logarithmic Sobolev Inequalities

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 {Px: 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 Jm: m1(E) → [0, ∞) ∪ {∞} is defined by: Obviously, (9.1) has interesting implications for the large deviation theory associated with {Px: x ∈ E}.