Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,π)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with $p< q$ implies a one-step entropy contraction estimate of the form ${\mathrm H}(μT\,|\,π)\le θ\, {\mathrm H}(μ\,|\,π)$, with $θ=p/q$. Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups.
Transition kernels. Throughout this note, we consider a measure-preserving transition kernel on a probability space (X, A, π), i.e. a map T : X × A → [0, 1] such that (i) A → T (x, A) is a probability measure for each x ∈ X;
(ii) x → T (x, A) is measurable for each A ∈ A;
(iii) π is fixed by the natural action of T on P(X), i.e. the map µ → µT given by (µT )(A) := X T (x, A) µ(dx).
(
Our aim is to shed a new light on the interplay between two fundamental regularization Hyper-contractivity. The transition kernel T naturally acts on non-negative measurable functions f : X → [0, ∞] via the familiar formula
This definition of course extends to signed functions by linearity, as long as T f + or T f - is finite. Moreover, for each p ≥ 1, Jensen’s inequality and the stationarity property πT = π easily and classically guarantee that the above action is a contraction on the Banach space L p (π), equipped with its usual norm,
Hyper-contractivity is the stronger requirement that, for some q > p and all f ∈ L p (µ),
The first estimates of this form were discovered by Nelson along the Ornstein-Uhlenbeck semigroup [12], and by Bonami and Beckner on the discrete hypercube [6,4]. Following the foundational contributions of Gross [9], Bakry and Émery [1], and Diaconis and Saloff-Coste [11], hypercontractivity has emerged as a fundamental tool in the quantitative study of Markov processes. In recent years, its impact has extended dramatically, yielding remarkable advances in statistical physics and computer science [10,3].
Entropy contraction. The second regularization property that we shall consider is a classical entropy contraction estimate, which takes the form
for some constant θ < 1. Here, H (• | •) denotes the Kullback-Leibler divergence:
Among several other applications, entropy contraction plays a prominent role in the analysis of mixing times of Markov processes [5,11], as well as in quantifying the celebrated concentration-of-measure phenomenon under the reference law π. We refer the unfamiliar reader to the recent lecture notes [7,13] and the references therein for a self-contained introduction, and many examples.
In the present note, we establish the following simple, general, and seemingly new quantitative relation between hyper-contractivity and entropy contraction.
Theorem 1 (Main result). For any parameters 1 ≤ p ≤ q, the hyper-contractivity estimate (3) implies the entropy contraction estimate (4), with θ = p/q.
We emphasize that Theorem 1 applies to any measure-preserving transition kernel on any probability space: neither reversibility, nor any sort of regularity is required.
In particular, we may think of T as the transition kernel of a discrete-time Markov process on X with invariant law π, in which case the conclusion can readily be iterated to provide a geometric rate of convergence to equilibrium, in relative entropy. Alternatively, we can choose T = P t , where (P t ) t≥0 is a given measure-preserving Markov semi-group on (X, A, π) and t ≥ 0 a particular time-scale which we want to investigate.
In this well-studied continuous-time setting, the ability to focus on a single instant appears to be new, and makes our static implication stronger than its celebrated dynamic counterpart, which we now review.
To lighten our discussion, we deliberately omit technical details and refer the interested reader to the excellent references [2] (for Markov diffusions on Euclidean spaces or smooth manifolds) and [8] (on finite state spaces). Consider a measure-preserving Markov semi-group (P t ) t≥0 on a probability space (X, A, π), and assume that it satisfies an exponential hyper-contractivity estimate of the form
for some β > 0. Then, a classical differentiation leads to the log-Sobolev inequality
where F is an appropriate class of probability densities on (X, A, π), and E(•, •) the Dirichlet form associated with the semi-group. Now, in view of the elementary estimate
the log-Sobolev inequality (6) always implies its “modified” version
which, by a Grönwall-type argument, finally guarantees the exponential entropy decay
In other words, the implication ( 5) =⇒ ( 8) always holds along Markov semi-groups.
This general relation between hyper-contractivity and entropy contraction is of course a well-established fact, with many applications. It is important to realize, however, that it is inherently dynamical: the estimates ( 5) and ( 8) hold for all t ≥ 0, and the semi-group structure is crucially used to reduce them to the respective functional inequalities ( 6) and (7), which can then be appropriately compared. In contrast, fixing a particular time t ≥ 0 and choosing T = P t in Theorem 1 directly yields
which, in view of the inequality 1 + e 4βt ≥ 2e βt , is always stronger than (8). On finite state spaces for example, this readily leads to the mixing-time estimate
which is twice better than what the traditional estimate (8) would give. Here, we have used the classical notation π ⋆ := min x∈X π
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