Logarithmic Sobolev Inequalities of Symmetric Diffusions

Abstract

Let E be a Polish space, M ₁(E) be the space of probability measures on E and B(E; ℝ) be the space of bounded measurable real valued functions on E. For a given μ ∈ M ₁ (E), we write $< f{ > _{\mu }} = {\smallint _{E}}fd\mu$ and $\left \| f \right \|_p=\left \| f \right \|_{Lp(\mu )}$. Let {P_t: t > 0} be a μ-symmetric Markov semigroup on E with generator L. We will suppose that the domain of L contains an algebra A $\subseteq$ B(E; ℝ) dense in L ^p(μ), 1 ≤ p < ∞, which is closed under L, P _t and the composition of C ^∞-functions. Moreover the semigroup should be sufficiently mixing so that for all f ∈ A,

$$ \mathop{{{\text{lim}}}}\limits_{{t \to \infty }} {P_{t}}f = < f > \mu $$

in

$$ L^p(\mu ),1\leq p< \infty $$

.